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DOCTOR  OF  PHILOSOPHY 


ISRAEL   EUCLID   RABINOVITCH 

April  29,  1901 


NEW   YORK: 
PUBLISHED   BY   THE  AUTHOR 

1903 


THE  FOUNDATIONS 


OF  THE 


EUCLIDIAN  GEOMETRY 


AS  VIEWED  FBOM  THE 


STANDPOINT  OF  KINEMATICS 


ERRATA. 

Page  X,  next  to  bottom  line  read  "  Bolyai ''  instead  of  Bolyi. 
Page  xi,  under  Lobatchevski,  close  the  quotation  with  ". 
Page  26,  last  line,  read  "  Matematiche "  instead  of  Mathe- 
matiche. 


^ET 


ISRAEL  EUCLID  RABINOVITCH 

Apeil  29,  1901 

OF  THE 

UNIVERSITY 

OF 

PUBLISHED   BY  THE  AUTHOR 
1903 


THE  FOUNDATIONS 


OP  THE 


EUCLIDIAN  GEOMETRY 


AS  VIEWED  FEOM  THE 


STANDPOINT  OF  KINEMATICS 


BISSERTATION 

SUBMITTKD    TO    THE 

BOARD  OF  UNIVERSITY  STUDIES 

OF    THE 

JOHNS   HOPKINS  UNIVERSITY 

IN   CONFOBMITY  WITH  THE  BEQUIBEMENTS  FOB  THE  DEGBEB  OF 

DOCTOR  OF  PHILOSOPHY 


BY 


ISRAEL   EUCLID   RABINOVITCH 

Apbil  29,  1901 


PUBLISHED   BY  THE  AUTHOR 
1903 


^* 


w^ 


Copyright  1903 
By  ISRAEL  EUCLID  RABINOVITCH 


All  rights  reserved 


PRtSIOF 

The  Nn  e«a  PRinTme  Conpaiv 

LAiCASTER,  Pa. 


SDetitcateti 


TO 

FABIAN  FEANKLIN,  Ph.D., 

FORMERLY  PROFESSOR  OP  MATHEMATICS  IN  THE  JOHNS  HOPKINS  UNIVERSITY, 

3rn  Grateful  ^cfenotoleiffment 

OF  BENEFITS  CONFEBRED  UPON  THE  AUTHOR  DURING  HIS   RESIDENCE  IN 

BALTIMORE  AS  A  GRADUATE  STUDENT  OF  THE 

JOHNS  HOPKINS  UNIVERSITY. 


lU 


166146 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/foundationsofeucOOrabirich 


PREFACE. 

The  present  work  is  the  result  of  long  meditations  and  of  an 
earnest  search  for  truth.  A  conviction  that  truth  in  mathe- 
matics must  be  absolute,  not  admitting  of  any  compromises, 
and  an  inmost  feeling  that  Nature  is  not  deceiving  us  and  that 
She  reveals  Herself  to  us  in  Her  true  appearance,  unmutilated 
by  false  logic,  have  guided  me  in  my  endeavors  to  solve  one  of 
the  hardest  mathematical  problems,  so  intimately  connected 
with  the  problem  of  the  origin  of  our  ideas,  —  namely,  the 
problem  of  the  Foundations  of  Geometry.  These  meditations 
were  finally  written  up,  in  April,  1899,  at  the  prompting  of 
my  excellent  and  highly  esteemed  friend  and  benefactor.  Dr. 
Fabian  Franklin,  formerly  Professor  of  Mathematics  in  the 
Johns  Hopkins  University,  to  whom  I  have  availed  myself  of 
the  present  opportunity  of  expressing  my  gratitude,  by  inscrib- 
ing this  work  to  him. 

Another  name  I  ought  to  mention  with  gratitude  is  that  of 
Dr.  Alexander  S.  Chessin,  Professor  of  Mathematics  in  Wash- 
ington University,  who  was  the  first  to  appreciate  the  value  of 
this  work  and  to  talk  to  me  unreservedly  about  it,  and  also  to 
urge  me  to  use  it  as  a  thesis  for  the  Ph.D.  degree. 

And,  finally,  I  owe  a  duty  of  gratefulness  to  my  distinguished 
professor.  Dr.  Frank  Morley,  for  his  guidance  in  the  work  of 
reading  up  the  literature  of  the  subject,  for  discussing  with  me 
points  of  difficulty  in  the  literature,  and  also  for  allowing  me  to 
present  some  of  my  theorems  before  the  conference  of  the 
mathematical  seminary  of  the  university,  where  the  general 
discussion  by  the  audience  helped  me  in  improving  the  mode 
of  presentation  of  these  theorems.  To  this  discussion  I  owe, 
in  particular,  the  analytical  presentation  of  my  proof  that,  with 
the  point  as  its  element,  space  must  be  three-dimensional.  The 
whole  of  the  Introduction  was  undertaken  and  carried  out,  dur- 
ing the  academic  year  of  1900-1901,  at  the  suggestion  and  under 
the  direction  of  Professor  Morley,  and  it  is  intended  as  a  critical 
review  of  some  of  the  most  important  results  obtained  by  modern 


VI  PREFACE. 

mathematicians  in  the  subject  (Riemann,  Beltrami,  Lie  and 
Poincar6),  so  that  in  the  light  of  these  an  adequate  estimate  of 
the  results  achieved  in  this  Dissertation  might  become  possible. 
And  last,  but  not  least,  my  thanks  are  due  to  Professor 
Edwin  R.  A.  Seligman  and  Professor  Felix  Adler  of  Columbia 
University  for  the  generous  interest  they  have  taken  in  the 
publication  of  my  work  in  full ;  and  also  to  Mr.  Isador  Goetz, 
A.B.,  of  New  York  City,  for  his  valuable  assistance  in  the  re- 
vision of  the  proof-sheets,  and  to  the  gentlemen  of  The  New 
Era  Printing  Company  for  the  care  and  efficiency  with  which 
the  printing  of  this  volume  has  been  executed. 

130  Heney  Street,  New  York  City, 
November,  1903. 


CONTENTS. 

Page. 
INTRODUCTION 1-37 

An  infinity  of  mutnally  contradictory  geometries  postulated  by 

Klein,  Killing,  and  others  —  and  its  criticism 1-5 

Sophns  Lie's  opinion  as  to  the  possibility  of  an  efficient  system 

of  postulates  and  axioms  of  geometry 5-6 

Lie's  treatment  of  the  Riemann-Helmholtz  space-problem  by 
the  theory  of  transformation-groups,  and  the  axioms  which 
he  assumes  for  this  purpose.  A  comparison  of  this  set  of 
axioms  with  those  assumed  in  the  Dissertation 6-9 

The  interpretation  of  the  non-Euclidian  groups  of  displacements 

represents  but  a  partially  solved  problem 10-11 

Poincare's  opinions  of  the  relation  of  pure  reason  and  experi- 
ence to  the  formation  of  our  geometrical  notions  ;  his  opin- 
ions as  to  the  number  of  dimensions  of  space 11-16 

An  account  of  Riemann's  Inaugural  Dissertation  on  the  Foun- 
dations of  geometry  and  multiply-extended  manifolds.  — 
Resume 16-20 

Importance  of  the  problem  from  a  purely  mathematical  point  of 
view.  Theory  of  proportion  and  similar  figures  impossible 
in  the  non-Euclidian  geometry  ;  the  squaring  of  the  circle 
possible 20-21 

A  summary  and  justification  of  the  author's  views  in  his  treat- 
ment of  the  subject.  The  conception  of  space  as  a  number 
manifold  of  three  dimensions  —  to  be  sought  in  the  properties 
of  rigidity,  impenetrability,  and  infinite  divisibility  of  mate- 
rial substance.  The  notion  of  distance  derived  from  rigidity. 
The  distance-line,  or  straight  line,  must  be  constructed  in 
space,  not  in  a  plane,  and  deduced  from  the  notion  of  dis- 
tance. The  plane  should  be  constructed  from  the  straight 
line,  and  both  should  be  demonstrated  to  have  all  properties 
commonly  attributed  to  them  without  justification.  For 
instance, — the  legitimacy  of  the  assumption,  of  the  plane 
being  capable  to  move  upon  itself  in  a  triply-infinite  number 
of  ways,  and  of  its  coincidence  with  itself  when  its  two  sides 
are  interchanged  —  must  follow  from  its  construction 21-25 

An  account  of  Beltrami's  views  of  the  interpretability  of  the 
geometry  of  Lobatchevski  upon  the  pseudosphere,  based  upon 
the  analogy  of  the  straight  line  and  a  geodesic  upon  that  sur- 
face, when  bending  without  stretching  is  allowed 26-27 

Distinction  emphasized  between  the  straight  line,  as  a  figure 
capable  of  lying  in  a  plane,  and  its  notion  as  a  geodesic  in  a 

vii 


Vlll 

Page. 
plane,  —  and  reason  given  why  in  the  later  case  the  Euclidian 

postulate  could  not  be  proved 27-29 

Quotations  from  Bianchi,  "  Lezioni  di  geometria  dififerenziale," 
concerning  the  futility  of  all  attempts  of  proving  the  Eu- 
clidian postulate,  and  a  justification  of  the  author's  attempt 
to  do  it  by  means  of  the  immaterial  quadrilateral,  not  bound 

to  lie  in  a  plane 29-32 

The  identity  of  the  author's  postulates  with  those  necessary  for 
defining  the  group  of  displacements  in  general.     A  comparison 

with  the  postulates  assumed  by  Helmholtz 32 

The  interpretation  of  the  non-Euclidian  geometries  again,  and 
Beltrami's  opinion  that  the  non-Euclidian  geometry  of  three 
dimensions  is  interpretable  only  analytically.  —  An  allusion 
to  a  probable  interpretation  in  line-space,  with  certain  special 
conventions  as  to  the  meaning  of  metrical  terms 32-37 

DISSERTATION. 

THE  FOUNDATIONS  OF  THE  EUCLIDIAN  GEOMETRY.     38-115 
CHAPTER  I. 
(inteoductoey.) 
Space  and  Its  Dimensions. 

Definitions  1-6  :  —  Geometry  ;  space,  extension  ;  impenetra- 
bility, geometrical  place  ;  vacant  space,  infinite  divisibility  ; 
magnitude  and  form  ;  geometrical  solid,  volume 38-39 

Postulate  1,  —  Motion  without  distortion.  Definitions  7-8  :  — 
Geometrical  equality,  coincidence  ;  non-equality.  Scholium 
and  R^sum6  :  —  A  rational  justification  of  the  definitions  and 
postulate 40-43 

Definitions  9-12  and  R^sum^:  —  Distance  and  contact  between 
bodies;  Surface  physical  and  geometrical;  Portions  and 
equality,  or  coincidence,  of  surfaces;  Homogeneous  and  non- 
homogeneous  surfaces 44-47 

Definitions  13-15:  — Line;  Portions,  equality  of  lines,  homoge- 
neous lines;  The  point 48-50 

Resume :  —  Discussion  of  the  tri-dimensionality  of  space 50-53 

Thirteen  paragraphs  purporting  to  give  an  analytical  discussion 
of  the  dimensions  of  space 54-62 

CHAPTER  IL 

The  Spheee,  Ciecle,  Steaight  Line,  Angle,  Teiangle,  Plane,  Etc. 

Definition  I, —  Distance.  Axiom  1, —  Continuity  in  congruence. 
Lemma  1,  — A  fixed  body 63-64 


IX 

Page. 
Theorem  1,  —  Construction  of  a  sphere.    Cor.  I, — Spherical 

surface  divides  all  space  into  two  regions.     Definition  II,  — 

Greater  and  smaller  distances.     Cor.  II, — Unique  center. 

Cor.  Ill,  —  Two  distinct  spheres  cannot  have  a  finite  portion 

of  surface  in  common 64-66 

Theorem  2:  —  Measurement  of  distances  from  a  fixed  origin; 
addition  and  subtraction  of  distances 67-72 

Theorems  3-5,  Operation  of  addition  of  distances  obeys  the 

associative  and  the  commutative  laws 72-78 

Scholium,  Definition  III  and  corollaries  I-IX:  —  The  homo- 
geneous distance-line,  or  straight  line,  and  its  properties 78-83 

Theorem  6,  Definition  IV:  —  Infinity  of  straight  line;  Angle. 
Theorem  7,  Definition  V:  —  A  right  angle  is  one  of  four  equal 
ones  formed  by  two  intersecting  straight  lines  in  space 84-85 

Theorem  8,  Definition  VI,  and  corollaries  I-V :  —  The  construc- 
tion of  a  plane  from  a  fixed  origin;  the  properties  of  a  plane 
as  an  infinite  homogeneous  surface  symmetrical  with  respect 
to  space.    Theorem  9, — All  right  angles  are  equal 85-89 

Scholiums  I  and  II:  —  Every  pair  of  crossing  lines  is  applicable 
to  a  pair  of  crossing  rays  in  a  plane;  measurement  of  angles 
by  arcs  of  the  circumference  of  a  circle. 

Theorem  10,  —  Adjacent  supplementary,  and  vertical  angles...      89-92 

Definition  VII  and  cor..  Theorem  11  and  cor.: — Simplest 
conditions  of  equality  of  triangles 92-93 

Theorem  12,  Definition  VIII  and  cor., — A  straight  line,  an 
angle,  and  a  triangle  are  plane  figures.  Scholium,  —  The 
segment  of  a  transversal  of  a  variable  angle  increases  with 
the  increase  of  the  angle 93-95 

Definition  IX,  Theorem  13  and  corollaries:  —  A  perpendicular 
to  a  plane.  A  plane  can  be  shifted  upon  itself  in  a  triply- 
infinite  number  of  ways,  and  also  turned  upside  down  with- 
out deformation 95-96 

CHAPTER  III. 

The  Quadbilateral,  the  Immaterial  Quadrilateral. 
Parallel  Straight  Lines. 

An  enumeration  of  the  theorems  of  the  first  book  of  Euclid 
which  can  easily  be  established  by  means  of  the  principles 
laid  down  earlier  in  this  Dissertation 97 

Theorem  14  and  corollaries  I-IV:  — Sum  of  two  interior  angles 
of  a  triangle  <2  rt.  /'s;  Two  straight  lines  making  with  a 
third  one  two  interior  ^'s  =  2rt.  Z's>  cannot  meet;  Exterior 
angle  of  a  triangle,  etc. ;  A  perpendicular  is  the  shortest  dis- 
tance from  a  point  to  a  line 98-99 


Page. 

Definition  X,  Theorems  15-16  and  corollaries, — The  quadrilat- 
eral, the  immaterial  quadrilateral,  and  some  of  their  proper- 
ties      99-101 

Theorem  17  and  cor. : — In  a  plane  quadrilateral  two  of  whose 
opposite  sides  are  equal,  and  in  which  two  interior  angles 
adjacent  to  the  third  side  are  supplementary,  this  third  side 
is  not  greater  than  the  one  opposite  to  it ;  The  sum  of  the 
three  angles  of  a  triangle  ^2rt.  ^'s 101-102 

Theorem  18,  —  The  possibility  of  a  certain  deformation  of  an 
immaterial  quadrilateral  with  equal  opposite  sides  established  102-106 

Theorem  19, — A  lemma  from  three-dimensional  geometry. 
Theorem  20  and  corollaries  :  —  The  motion  considered  in 
Theorem  18  is  a  plane  motion  ;  Conclusion  as  to  the  sum  of 
the  interior  angles  adjacent  to  the  same  side  in  an  immaterial 
quadrilateral  subject  to  such  a  motion 106-108 

Theorem  21  and  corollaries  : — The  sum  of  the  interior  angles 
adjacent  to  the  same  side  of  a  plane  quadrilateral  with  equal 
opposite  sides,  equals  two  right  angles  ;  The  sum  of  the  three 
angles  of  a  triangle,  and  the  sum  of  the  four  angles  of  any 
plane  quadrilateral 108 

Theorem  22, — Properties  of  two  lines  perpendicular  to  a  third  in 
the  same  plane 108-110 

Definition  XI, — Provisional  definition  of  parallel  lines  and  cor- 
ollaries   110-111 

Theorem  23  and  corollaries  :  —  Three  collinear  points  at  equal 
distances  from  a  given  straight  line,  must  be  in  the  same  plane 
with  the  latter, — and  a  consequent  further  definition  of  par- 
allel lines 111-113 

Theorem  24, —  A  proof  of  Euclid's  Eleventh  Axiom,  and  corol- 
laries extending  the  definition  of  parallels  to  that  given  by 
Euclid 113-115 

Conclusion 115 

Autobiography 116 

LIST  OF  WORKS  QUOTED  IN  THE  INTRODUCTION  OR  CONSULTED 
BY  THE  AUTHOR  IN  PREPARING  THE  DISSERTATION. 

Ball,  R.  S.,  —  "Measurement,"  article  Encyclop.  Brit.,  vol.  XV. 

Beltrami: — "Saggio  di  interpretazione  della  geometria  non-Euclidea, " 
Giornale  di  Matematiche,  1868,  t.  VI  ;  "Teoria  fondamentale  degli 
spazii  di  curvatura  costante, "  Annali  di  Matematica,  Ser.  2,  II,  1868. 

Bianchi,  —  "Lezioni  di  geometria  difEerenziale,"  German  translation,  1898, 

t.  II. 
Bolyi,  John,  —  "Science  Absolute  of  Space,"  translated  into  English  by 

Professor  George  Bruce  Halstead. 


XI 

Cantor,  —  Mathematisohe  Annalen,  t.  V,  pp.  123-128. 

Chrystal,  Geo.,— "Parallels,"  article  Ed  cyclop.  Brit.,  vol.  XVIII. 

Clifford,  —  Articles  on  the  Axioms  of  Geometry  in  his  Mathematical  Papers. 

Frischauf,  —  "  Eleraente  der  absoluten  Geometric,"  1876. 

Helmholtz, — "  Ueber  die  Thatsachen  die  der  Geometric  zum  Grunde  liegen," 
Konigliche  Gesellsohaft  der  Wissenschaften  zu  Gottingen. 

Henrici,  CI.,  —  *'  Geometry,"  article  Encyclop.  Brit.,  vol.  X. 

Killing,  —  "Ueber  die  Grundlagen  der  Geometric,"  Crelle's  Journal,  Bd. 
86,  109. 

Klein,  Felix:  —  Memoirs,  " Zur  Nicht-Euklidischen  Geometric,"  Mathe- 
matisohe Annalen,  Bd.  IV,  VI,  XXXVII;  "  Nicht-Euklidische  Geo- 
metric," lithographed  lectures,  Gottingen,  1893. 

Lie,  Sophus:  —  " Transformationsgruppen, "  vol.  Ill;  "  Continuierliche 
Gruppen  "  ;  "  Die  Grundlagen  der  Geometric,"  Leipziger  Beriohte. 

Lindemann,  —  Clebsch,  "Vorlesungen  iiber  Geometric,"  Bd.  II. 

Lobatchevski :  —  "Theory  of  Parallels,"  translated  by  Halstead  ;  F.  EngePs 
' '  Urkunden  zur  Geschichtc  der  Nicht-Euklidischen  Geometric. 

Playfair's  Euclid. 

Poincare :  —  Memoirs  on  the  foundations  of  geometry  in  the  :  Bulletin  de  la 
Soci^t^  Math^matiquc  de  France,  t.  16,  1887  ;  Revue  Generale  des 
Sciences  Pures  et  Appliquees  ;  Revue  de  M^taphysique  et  de  Morale  ; 
"The  Foundations  of  Geometry,"  Monist,  vol.  IX,  1888,— translated 
into  English  by  McCormack. 

Riemann,  —  "Ueber  die  Hypothesen  wclche  der  Geometric  zu  Grunde 
liegen,"  Mathematisohe  Werkc,  pp.  254-269. 


INTRODUCTION. 


A   SURVEY  OF   THE    MOST   IMPORTANT  VIEWS   OF    MODERN  MATHE- 
MATICIANS  ON   THE   FOUNDATIONS    OF    GEOMETRY. 


Both  mathematicians  and  philosophers  at  present  agree  that  — 
although  the  science  of  mathematics  as  a  whole  is  undoubtedly 
the  most  exact  of  sciences,  one  of  her  most  important  branches, 
at  once  the  oldest  and  the  most  fascinating,  namely  geome- 
try, has  to  some  extent  lost  in  its  prestige  and  can  no  longer  be 
quoted  by  epistemologists  as  the  prototype  of  purely  deductive, 
a  priori  science,  and  as  a  proof  of  the  existence  of  certain  in- 
nate ideas,  having  a  purely  transcendental  origin,  wholly  inde- 
pendent of  experience  and  partly  conditioning  it.  This  change 
of  view  upon  geometry  has  taken  place  during  the  past  cen- 
tury of  all-pervading  doubt  and  criticism,  and,  strange  to  say, 
it  did  not  originate  with  men  outside  the  profession  of  mathe- 
matics, but  with  those  who  had  the  greatest  interest  in  pre- 
serving her  sanctity,  in  keeping  up  the  halo  of  her  alleged 
transcendental  origin.  The  very  priests  who  worship  at  her 
shrine,  the  greatest  mathematicians  who  contributed  most  to 
her  fabulous  growth  and  development  in  the  nineteenth  cen- 
tury, —  Gauss,  Riemann,  Helmholtz,  Beltrami,  and  Clifford 
among  the  immortal  dead,  and  many  prominent  names  still 
among  us,  —  have  done  most  to  cause  this  change.  At  present, 
it  is  almost  regarded  as  a  heresy  to  attempt  to  restore  some  of 
the  old  prestige  to  the  science  which  was  considered  by  the 
Greeks  to  be  the  prototype  of  all  science  and  all  philosophy. 

All  this  change  of  view  has  occurred,  of  course,  not  with  re- 
gard to  the  method  employed  by  geometry,  the  soundness  and 
legitimacy  of  which  have  never  been  seriously  doubted,  but  with 
regard  to  the  very  foundations  upon  which  geometry  rests.  It 
is  the  body  of  axioms  and  postulates,  the  definitions  and  common 
notions,  —  propositions  and  assumptions,  both  implicit  and  ex- 

1 


plicit,  which  had,  for  a  very  long  time,  been  considered  as  self- 
evident,  intuitive,  and  independent  of  all  elaborate  proof, — prop- 
ositions that  need  only  be  stated  in  order  to  elicit  unconditional 
consent,  —  it  is  this  body  of,  so-called,  self-evident  truths  which 
at  present  are  questioned  and  doubted,  and  by  many  relegated 
to  the  realm  of  empiricism,  true  only  with  a  certain  degree  of 
approximation,  and  capable  of  being  modified  in  an  infinity  of 
ways  and,  hence,  of  giving  rise  to  a  corresponding  multitude  of 
geometries,  each  consistent  in  itself  but  in  contradiction  with 
the  others,  each  as  perfect  in  theory  as  any  other,  and  all  very 
nearly  agreeing  with  our  limited  experience.*  It  is,  however, 
admitted  on  all  sides,  that  the  old  system,  namely,  the  Euclidian 
system,  more  than  all  others,  seems  to  agree  with  the  results  of 
our  experience,  as  far  as  this  last  goes ;  and  if  we  were  able  to 
extend  our  experience  considerably  beyond  the  limits  of  the 
fixed  stars,  and  if  even  then  we  should  find  its  norms  to  remain 
unaltered  and  not  needing  revision,  it  would  to  a  certain  extent 
prove  the  physical  reality  of  the  Euclidian  geometry  and  the 
unreality  of  the  other  systems,  although  the  others  would  still 
be  theoretically  admissible  and  would  form  a  body  of  imaginary 
geometries. 

Now,  this  is  rather  a  peculiar  state  of  the  geometrical  science, 
singular  in  its  kind.  For,  while  in  other  branches  of  science 
two  contradictory  systems  of  thought  would  hardly  be  allowed 
to  stand  side  by  side,  both  claiming  to  represent  the  truth  simul- 
taneously, —  while,  for  instance,  the  Ptolemaic  and  Copernican 
systems  of  astronomy  could  not  avowedly  coexist  —  the  latter 
having  superseded  the  former  as  soon  as  it  was  found  to  agree 
better  with  astronomical  observations  and  with  the  abstract 
laws  of  mechanics,  —  while  no  quarter  was  given  to  the  Aristo- 
telian theory  of  the  fixity  of  species  by  the  new  evolutionary 
systems  of  Lamarck  and  Darwin,  or  to  the  ancient  doctrine 
of  the  Four  Elements  by  modern  chemistry  and  physics,  —  the 
contradictory  systems  of  geometry,  according  to  some  mathe- 
maticians of  note,  could  be  allowed  to  stand  together,  side  by 
side,  and  be  of  equal  theoretical  (if  not  practical)  value  and 
importance.     So,  for  instance,  F.  Klein,  in  his  memoirs  in  the 

*  Professor  F.  Klein  in  many  places  in  his  memoirs  on  the  non-Euclidian 
geometry,  Math.  Ann.,  Bd.  IV,  VI,  XXXVII,  and  in  his  ''Nicht-Euklidische 
Geometric,"  lithographed  impression,  Gott.,  1893,  forcibly  presents  and  de- 
fends this  opinion.  See  lithogr,  lectures,  I,  pp.  298-365;  Math.  Ann., 
XXXVII,  p.  570. 


Mathematische  Annalerij  vols.  4,  6,  7  and  37,  and  in  his  "  Nicht- 
Euklidische  Geometrie,"  second  impression,  Gottingen,  1893, 
develops  [from  the  projective  point  of  view  three  systems  of 
geometry,  —  the  Elliptic,  the  Hyperbolic,  and  the  Parabolic 
systems  (which  in  broad  features  had  been  drawn  already  by 
Riemann),  corresponding  to  the  three  possible  hypotheses  which 
can  be  made  concerning  our  space,  namely,  as  possessing  con- 
stant positive,  negative,  or  zero,  curvature. 

Giving  no  theoretical  preference  to  any  of  these  systems,  he 
even  goes  so  far  as  to  think  that  the  question,  whether  one 
of  these  systems  is  to  be  preferred  as  expressing  the  real  rela- 
tions of  our  space,  is  unanswerable,  since  by  allowing  the 
radius  of  curvature  to  be  sufficiently  great,  the  elliptic  or  hy- 
perbolic geometry  would  give  results  approximating,  with  as 
great  a  degree  of  accuracy  as  we  please,  to  the  results  obtain- 
able by  the  most  exact  measurements,  performed  with  the  most 
powerful  telescopes  upon  distances  such  as  are  involved  in 
the  determination  of  the  annual  parallax  of  a  fixed  star.  He 
prefers  the  parabolic  geometry,  however,  on  account  of  its  pre- 
senting the  simplest  hypothesis  in  the  theory  of  measurement. 
So  he  says  in  his  lectures  on  the  non-Euclidian  geometry, 
referred  to  above,  first  part,  page  277:*  "There  is,  how- 
ever, on  the  other  hand,  no  lack  of  enthusiasts,  who  do  not 
answer  the  question  in  the  way  we  have  done,  by  asserting  that 
to  our  conception  and  experience  of  space  could  with  sufficient 
precision  correspond  alike  the  hyperbolic  or  the  elliptic,  as  well  as 
the  parabolic  system  of  measurement,  and  that  we  decide  in 
favor  of  the  parabolic,  solely  on  account  of  its  offering  the 
simplest  hypothesis  (just  as  in  physics,  among  hypotheses  of 
equal  probability,  the  simplest  is  always  allowed  to  prevail)." 
Each  of  these  geometries,  according  to  Klein,  in  another  place  in 
the  same  work  (p.  295),  admits  of  an  infinity  of  different  space- 
forms,  —  in  which  respect  he  differs  from  Killing,  who  thinks 
that  only  in  case  of  the  elliptic  geometry  an  infinite  variety  of 
space-forms  —  Raumformen  —  is  possible.  He  says :  "  We  thus 
expressly  contradict  the  remark  of  Killing  that  in  case  of  the 
hyperbolic  or  parabolic  metrics  there  exists  the  possibility  only 
of  one  space-form ;  we  say,  on  the  contrary,  that  also  in  these 
cases  there  exists  an  infinity  of  space-forms."  —  It  has  come  to 

*  See  also  pp.  161-170  of  same  work,  where  this  idea  is  presented  with 
especial  force  and  elegance. 


pass,  indeed,  that  some  mathematicians  are  vying  with  one 
another  in  devising  new  space-forms,  as  they  call  them,  for 
which  new  systems  of  geometry  are  supposed  to  hold.  Clif- 
ford, Klein,  Lindemann,  Killing,  and  others  have  contributed 
much  to  this  field  of  investigation.  The  enumeration  and  de- 
scription of  some  of  these  geometries  would  lead  us  too  far, 
and  the  reader  interested  is  referred  to  the  numerous  memoirs 
in  the  Math.  Ann.  and  in  Crelle^s  Journal  and  to  separate 
books  and  reprints  from  mathematical  and  philosophical  peri- 
odicals, which  have  appeared  since  the  beginning  of  the  seven- 
ties up  to  the  end  of  the  past  century. 

It  may,  however,  be  said  without  exaggeration,  that  most  of 
these  space-forms  impress  the  reader  rather  with  the  ingenuity 
of  their  inventors  than  with  their  actual  value  in  bearing  upon 
the  question  of  the  foundations  of  geometry.  There  seems  to 
be  rather  too  much  license  given  to  the  imaginative  faculty  of 
the  human  mind ;  and  while  the  origin  of  almost  all  investiga- 
tions of  this  nature  is  to  be  sought  in  the  impetus  given  to  the 
non-Euclidian  geometry  by  the  deep-searching  criticism  of 
Riemann's  inaugural  dissertation  on  the  foundations  of  geome- 
try,*—  where,  it  may  be  said,  he  formulated  questions  without 
giving  final  answers  to  some  of  them, — the  new  systems  devised 
hardly  ever  carry  conviction  with  them,  and  it  may  be  stated 
as  a  certainty,  that  many  of  them  would  not  stand  a  scrutiniz- 
ing criticism  and  would  have  to  be  relegated  to  the  realm  of 
fancy  rather  than  be  classed  with  such  an  exact  science  as 
mathematics.  To  quote  an  instance,  the  elliptic  space,  i.  e.,  one 
of  positive  curvature  with  two  geodesies  meeting  only  in  one 
point,  described  by  Klein,  Lindemann  f  and  Killing, |  is  one  of 
such  systems.  Beltrami  in  his  "Teoria  fondamentale  degli 
spazii  di  curvatura  costante  '^  §  makes  the  express  statement 
that  any  two  geodesies  in  a  space  of  positive  curvature  meet  in 
two  antipodal  points,  through  which  a  whole  pencil  of  (an 

*"Ueber  die  Hypothesen,  welche  der  Geometrie  zu  Grunde  liegen," 
Math.  Werke,  pp.  254-269.  It  was  not  intended  for  publication  by  the 
author  in  the  form  it  appeared  after  his  death  in  the  Gottinger  Abhand- 
lungen.  See  "Nioht-Euklid.,"  I,  p.  206  ;  Lie,  "  Transformationsgruppen, " 
III,  pp.  485-486. 

t  Clebsoh,  Vorlesungen  iiber  Geometrie,  t.  II. 

X  Crelle,  t.  86,  p.  72. 

\AnnnH  di  Matematica,  ser.  2,  II,  1868  (The  French  translation  of  this 
work  of  Beltrami  and  also  of  his  famous  *'Saggio,"  by  Hoiiel,  appeared  in 
the  Jour,  de  VEcole  Normale,  t.  VI). 


infinity  of)  similar  geodesies  must  pass ;  and  to  this  statement 
Klein  takes  exception  in  his  "  Nicht-Euklid.  Geometric,"  pp. 
240-242  and  in  other  places  of  the  same  work,  and  in  the  Math, 
Ann.,  t.  6,  p.  125  and  t.  37,  p.  554  et  seq.  Another  instance 
is  the  spiral  space-form,  in  which  a  rotation  of  a  rigid  body  is 
accompanied  by  an  increase  or  decrease  in  its  volume,  so  that 
by  a  continuous  rotation  about  a  fixed  point,  any  arbitrarily 
chosen  body  can  be  made  to  enclose  any  given  point  of  space, 
and  by  an  inverse  rotation  the  body  can  be  made  to  shrink 
down  to  an  arbitrarily  small  portion  of  space  around  the  fixed 
point.  (Killing,  "  Ueber  die  Grundlagen  der  Geometric," 
Crelle's  Jour.,  t.  109,  1891,  pp.  185-266). 

There  are,  however,  other  mathematicians,  who, — agreeing  in 
the  main  that  the  foundations  of  geometry  have  thus  far  not 
been  laid  down  with  any  degree  of  certitude  and  that  they  are, 
therefore,  open  to  considerable  differences  of  opinion,  —  think, 
nevertheless,  that  there  ought  to  be  some  objective  truth  con- 
cerning the  nature  of  these  foundations,  and  that  it  is  not  at  all 
unlikely  that  some  day  a  satisfactory  body  of  axioms  and  pos- 
tulates may  be  found,  which  will  prove  undebatable.  The 
question  is  only  in  finding  the  minimum  of  simple  truths,  de- 
rived from  experience  as  an  original  source  and  formulated  by 
abstraction  into  a  body  of  definitions  and  propositions  which, 
—  on  account  of  their  incontestable  efficiency  as  a  basis  for 
geometry,  on  the  one  hand,  and  by  their  unquestionable  real- 
ity, on  the  other,  as  well  as  by  their  being  irreducible  to  a 
smaller  number  with  equal  efficiency,  —  should  carry  conviction 
into  the  mind  of  the  mathematician,  whose  taste  is  especially 
fastidious  in  this  relation,  and  should  satisfy  him  that  the  basis 
is  a  unified  whole,  without  leaks,  and  that  it  is  capable  of  stand- 
ing the  test  of  a  scientific  scepticism  (of  course,  not  a  meta- 
physical scepticism  putting  questions  of  the  nature  of  whether 
space  and  time  or  even  matter  and  mind,  the  ego,  the  universe, 
and  so  on,  have  any  reality,  objective  or  subjective,  phenomenal 
or  noumenal,  etc.).  Among  these  mathematicians  is  especially 
to  be  mentioned  Sophus  Lie,  who,  it  seems,  has  contributed 
more  than  any  contemporary  mathematician  to  sound  views  in 
this  matter,  by  treating  the  so-called  Riemann-Helmholtz  prob- 
lem in  a  masterly  way,  which  won  for  him  the  Lobatchevski 
Prize  in  1897.  So  Lie  says  in  his  "  Transformationsgruppen," 
Vol.  Ill,  p.  398,  "  We  wish,  however,  to  express  the  opinion, 


6 

which  is  a  conviction  with  us  (wollen  wir  doch  als  unsere  Ueber- 
zeugung  die  Auffassung  aussprechen),  that  it  is  in  no  wise  im- 
possible to  establish  a  system  of  geometrical  axioms  which  at  once 
shall  be  sufficient  and  shall  contain  nothing  superfluous.  It  is 
distressingly  certain,  however,  that  there  are  very  few  investiga- 
tions which  have  actually  furthered  the  problem  as  to  the  foun- 
dations of  geometry." 

In  another  place  in  the  same  work  (p.  536)  he  says  :  "  Ge- 
ometry in  its  different  stages  ought,  as  much  as  possible,  to  be 
founded  on  a  purely  geometrical  basis ;  this  is  a  demand  with 
which  everybody  undoubtedly  will  agree. 

"  For  the  first  stage  of  geometry  are  necessary,  in  the  first 
place,  certain  fundamental  conceptions,  like  space,  curve,  point, 
and  surface ;  second  come  certain  axioms  concerning,  for  in- 
stance, the  properties  of  the  right  line,  the  existence  of  a  sphere, 
and  so  on.  Every  new  conceivable  stage  is  characterized  by 
the  introduction  of  new  axioms, — one  stage,  for  instance,  by  the 
axiom  of  parallels,  another  by  the  Cantor-axiom." — In  my  own 
work,  this  last  axiom,  i.  e.,  that  the  straight  line  is  a  number- 
manifold,  will  be  proved  to  be  one  of  the  fundamental  properties 
of  the  straight  line,  from  which  its  construction  and  all  its  other 
properties  are  obtained. — "  Upon  this  axiom  it  is  possible  to 
establish  rationally  the  conceptions  of  area,  length  of  arc,  etc., 
while  Euclid  virtually  needs  a  separate  axiom  in  each  case. 

"  The  great  question  is  now,  what  axioms  in  each  stage  are 
not  only  sufficient  but  also  necessary,  in  other  words,  are  in- 
dispensable. In  the  answer  to  this  question  the  whole  problem 
of  the  foundations  of  geometry  would  find  its  solution. 

And  then  further  (p.  537)  Lie  sketches  a  programme  for  the 
mathematician  who  would  undertake  to  establish  the  necessary 
axioms,  which  perhaps  might  prove  fewer  in  number  than  those 
which  Lie  assumed  provisionally  for  the  purpose  of  solving  the 
Riemann-Helmholtz  problem. — "  First  one  would  have  to  estab- 
lish certain  fundamental  conceptions,  like  space,  point,  curve, 
surface,  and  also  the  conception  of  motion.  .  .  . 

"  As  a  first  axiom  one  would  have  to  establish  the  following  : 
If  a  point  P  is  fixed,  every  other  point  can  still  describe  a  sur- 
face which  does  not  pass  through  the  point  F.  In  this  may  be 
found  the  reason  that  two  points  in  all  rigid  motions  (Bewegun- 
gen)  remain  separated."  [These  two  conceptions  are  certainly 
also  connected  in  my  treatment  (see  pp.  63-64,  definition  of 


distance  and  the  proof  following  of  the  existence  of  a  sphere), 
except  that  the  second  is  coming  first,  as  the  simplest  one,  and 
following  at  once  from  the  conception  of  rigidity.] 

"  As  a  second  axiom  it  would  be  necessary  to  assume  that,  — 
When  two  points  P^  and  P^  are  fixed,  there  are  still  an  infinity 
of  other  points  which  remain  fixed  simultaneously,  and  these 
points  form  one  and  only  one  line  passing  through  Pj  and  Pg." — 
I  think  Lie  would  certainly  not  object  to  a  proof  of  this  prop- 
osition, which  at  once  makes  space  a  number-manifold  and, 
thus,  supplies  also  the  Cantor-axiom.  He  says  that  these  prop- 
ositions are  not  yet  sufficient  for  the  first  stage  of  geometry. 
Now,  I  think,  that  by  means  of  only  one  additional  axiom  (viz, 
axiom  1,  p.  63),  which,  according  to  my  mind,  happens  to 
coincide  with  Lie's  fundamental  notion  of  a  continuous  group 
of  displacements,  I  have  succeeded  in  establishing  all  that  is 
necessary  for  the  first  stage  together  with  the  most  important 
postulate  of  the  second  stage,  namely,  the  postulate  of  parallels, 
— in  a  sense,  however,  that  does  not  exclude  the  spherical  and 
pseudospherical  geometries,  proving  only  the  necessity  of  a 
plane  geometry  in  the  Euclidian  sense. 

Lie  himself  treated  this  subject  from  the  point  of  view  of 
continuous  group-transformations.  Starting  with  the  Riemann- 
Helmholtz  postulate  that  space  is  a  manifold  of  three  dimen- 
sions, in  which  the  position  of  the  single  element,  the  point,  is 
determined  by  three  coordinates,  and  adding  a  few  very  simple 
postulates,  characterizing  the  group  of  continuous  motions  of 
rigid  bodies  (i,  e.,  transformations  in  which  every  two  points 
have  one  essential  invariant),  he  showed  that  there  remains 
only  the  possibility  of  the  Euclidian  and  the  two  non-Eucli- 
dian systems  of  motions.*  The  latter  two  are  such  as  leave 
invariant  respectively  the  imaginary  surface  x?  -f-  icl  +  icf  -f 
1  =  0  (Eiemannian  group),  or  the  real  surface  xl  -{-  xl -}-  x^ -- 
1  =  0  (Lobatchevski  group  of  motions).  The  group  of  Eucli- 
dian motions  together  with  the  group  of  transformations  by 
similar  figures  are  characterized  by  their  leaving  invariant  the 
absolute, 

x'  +  f  +  z'^O,     <=0.t 

*  Transformationsgruppen,  III,  pp.  464-479.  The  whole  of  the  fifth 
chapter  is  devoted  to  the  Riemann-Helmholtz  problem.  See  also  Leipziger 
Benchte,  1890,  pp.  356-418,  284-321,  and  1892,  pp.  297-305. 

t  TransformatioDsgruppen,  III,  p.  218. 


8 

The  following  are  the  axioms  *  which  Lie  considers  suffi- 
cient for  his  investigation  : 

I.  Space  is  a  number-manifold  of  three  dimensions,  R^, 

II.  The  displacements  or  motions  of  B^  form  a  real  continu- 
ous group  of  point-transformations. 

III.  If  any  real  point  of  general  position,  3/ J,  y^,  3/3,  is  held 
fixed,  all  the  real  points  x^^  x^y  x^  into  which  another  real  point 
tcj,  xlf  xl  can  be  moved,  satisfy  an  equation  with  real  coeffi- 
cients, of  the  form 

^{Vv  2/2?  Vs  »    ^V  ^2>  ^3  J    ^V  ^2'  ^3)  ~  ^f 


which  is  not  fulfilled  for  0;^  =  ^J,  X2  =  yl,  x^  =  1/3,  and  which, 
in  general,  represents  a  real  surface  passing  through  x^,  x^^  xl. 
(A  synthetic  proof  of  this  proposition  is  given  in  Theorem  1, 
p.  64,  of  my  Dissertation.) 

IV.  About  the  point  2/J,  yl,  yl  a  finite  triply-extended  region 
may  be  so  bounded  that,  after  fixing  the  point  y^,  y\^  yl,  every 
other  real  point  x\j  x\,  xl  of  the  region  can  still  pass  by  con- 
tinuous motion  into  the  position  of  every  other  real  point  of  the 
region  which  satisfies  the  equation,  TF=  0,  and  which  is  joined 
to  the  point  y\y  yl,  yl  by  an  irreducible  continuous  series  of 
points.  (This  proposition  is  also  proved  in  the  theorem  referred 
to  above.) 

In  the  Leipziger  Berichte,  1890,  pp.  357-358,  he  character- 
izes the  axioms  necessary  and  sufficient  to  define  the  Euclidian 
and  the  two  kinds  of  non-Euclidian  motions  in  a  somewhat 
different  way,  which,  however,  amounts  to  the  same  thing. 
The  analytical  formulae  are  interesting,  and  I  repeat  them  here. 
The  assumptions  are  : — 

An  infinite  aggregate  of  real  transformations  of  the  points  ol 
JR3  (xy  y,  z)  is  given  by  the  equations  : — 

^1  =/(^>  y>  h  «!>  «2?  •  •  •)>   Vi  =  *(^?  y^  h  ^v  ^v  ' '  ')> 

z^^y\r{x,y,z,a^,a,^,  . . .). 

These  equations  have  to  satisfy  the  following  conditions  : 
A.  The  functions  /,  </>,  -^  are  analytical  functions  of  the  co- 
ordinates X,  y,  z  and  of  the  parameters  a^,  a^,  a^,  ■  •  • 

*  J64U,  pp.  506-507. 


B.  Any  two  points  x^^  y^,  z, ;  x^^  y^,  z^  have  one  essential 
invariant  under  all  transformations  of  the  group,  of  the  form 

^{^v  Vv  \  ^  ^2^  Vv  h)  =  const. 
Hence, 

^(^v  Vv  h  >  ^21  2/2?  ^2)  =  ^«y  y'v  <  y  ^'v  Vv  ^2% 

where  x[j  y[,  z[ ;  x'^,  y'^,  z'^  are  the  new  positions  of  the  original 
pair  of  points  x^,  y^,  z^ ;  x^,  y^,  z^,  which  these  obtain  in  virtue 
of  any  transformation  of  the  group. 

C.  The  group  is  transitive,  i.  e.,  any  point  of  the  -R3  can  be 
transformed  into  any  other  point.  If,  however,  one  point 
x^,  y^y  z^  is  iixed,  every  other  point  x.^,  y^,  z^  can  assume  00^ 
different  positions,  which  are  defined  by  the  equation  : 

^(^1,  yv  h ;  ^v  Vv  0  =  ^{^v  Vv  ^i  y  ^v  Vi^  ^2)' 

If  two  points  x^f  y^y  z^  and  x^y  y^y  z^  are  fixed,  any  third  point 
of  general  position,  x^y  3/3,  z^y  can  assume  00^  different  positions 
SC3,  y'^y  z'^  defined  by  the  equations : 

^  (^1?  Vv  ^i  >  ^3^  2/3J  h)  =  ^  (^v  Vv  h  y  ^v  Vv  ^3) 

^  (^2>  Vv  \ ;  %y  y'v  ^D  =  ^  (^2>  yt>  ^  >  ^3>  2/3^  ^3)- 

[The  point  x^y  y^y  z.^  must  be  of  general  position,  in  order  to  be 
able  to  assume  oo^  different  positions,  since  there  exists  a  singly- 
infinite  number  of  points  defined  by  the  last  two  equations  II, 
such  that  x'^y  2/3,  z[^  =  x^y  y^y  Z3,  namely  the  00^  points  collinear 
with  x^y  y^y  z^ ;  x^,  y^y  z^.]  ^  If  three  points  x,^y  y^y  2, ;  x^y  y^,  z^ ; 
x^y  3/3,  z^  are  fixed,  all  points  of  space  remain  fixed,  and  three 
similar  equations  will  be  satisfied  only  for  x'^  —  x^y  y'^  —  y^y  z\ 


The  interpretation  of  the  results  obtained  by  Lie  with  regard 
to  the  possibility  of  the  two  groups  of  non-Euclidian  motions, 
according  to  my  mind,  represents  still  an  unsolved  mathemat- 
ical problem,  alongside  with  the  interpretation  of  similar  re- 
sults obtained  by  him  for  n-dimensional  manifolds.  A  concrete 
interpretation  of  these  must  be  found  in  our  empirical  space, 
for  which  the  Euclidian  axioms  hold,  in  order  to  appreciate 
their  full  geometrical  significance. 


10 

Not  considering  myself  competent  at  present  to  undertake  even 
a  partial  solution  of  the  problem,  I  wish,  however,  to  indicate 
that  for  two-dimensional  manifolds  an  interpretation  seems  to 
be  near  at  hand,  not  at  all  in  conflict  with  our  Euclidian  con- 
ceptions of  space.  The  non-Euclidian  groups  of  transforma- 
tions, assuming  an  invariant  between  two  elements,  only  of 
the  most  general  kind,  and  presenting  projective  relations  to 
certain  special  forms  of  the  fundamental  quadric,  must  repre- 
sent the  real  metric  relations  of  our  space,  as  imaged  by  pro- 
jection upon  certain  surfaces  of  the  second  degree.  In  such  a 
projective  image  the  anharmonic  ratio  of  four  points,  or  some 
function  of  it,  will  remain  unaltered,  which  will  have  to  be 
taken  for  a  definition  of  distance  or  angle  in  these  transformed 
metrics;  so  that  these  metrics  will  coincide  with  the  generalized 
Cayleyan  metrics,  developed  by  Klein.  For  an  interpretation 
of  this  nature,  we  may  refer  to  Poincar^'s  paper  on  the  founda- 
tions of  geometry  in  the  Bull,  de  la  Soc.  Math,  de  France,  t.  16, 
Nov.,  1887.  Another  interpretation  for  these  groups  has  been 
found  in  the  metrics  upon  surfaces  of  positive  and  negative 
curvature,  when  the  straight  lines  are  replaced  by  geodesies  on 
these  surfaces,  —  an  interpretation  which  has  been  fully  justi- 
fied by  the  works  of  Beltrami,  to  which  reference  will  be  made 
later.  Klein,  in  his  "  Nicht-Euklid.  Geometric,'^  has  shown,  it 
seems  to  me,  conclusively  (although  he  intended  his  procedure 
to  illustrate  and  to  show  in  a  concrete  manner  the  possibility 
of  the  plane  having  elliptic  or  hyperbolic  metrics)  that  what  he 
calls  an  elliptic  plane  is  actually  the  central  projection  upon  a 
Euclidian  plane  of  the  metrics  upon  a  sphere,*  and  what  he 
calls  the  hyperbolic  plane  is  the  orthographic  projection  of  a 
system  of  metrics  upon  a  sphere  touching  the  plane,  in  which 
the  straight  lines  are  represented  by  circles  orthogonal  to  the 
equator  of  the  sphere  which  is  parallel  to  the  plane.  At  any 
rate,  this  also  shows  that  by  certain  processes  of  projection  and 
by  making  certain  conventions  as  to  the  meaning  of  "  distance  " 
and  "angle,''  we  may  be  able  to  account  for  the  two  non- 
Euclidian  groups  of  motions,  as  well  as  for  the  non-Euclidian 
metrics  deduced  by  Klein  from  the  Cayleyan  metrics.  The 
non-Euclidian  groups  of  displacements  for  three  dimensions, 

*  The  radius  of  the  sphere  is  taken  to  be  =  2k,  and  ±  l/4fc2  is  the  measure 
of  curvature  of  the  elliptic  or  hyperbolic  plane,  resp.  See  "  Nicht- 
Euklid.,"  pp.  94-97  and  pp.  220-237. 


11 

however,  as  well  as  the  Euclidian  and  the  non-Euclidian 
groups  of  displacements  for  manifolds  of  a  higher  number  of 
dimensions  than  three,  which  Lie  derives  from  postulates  simi- 
lar to  those  he  assumed  for  R^,  still  need  an  interpretation,  and 
this  interpretation  seems  to  lie  in  the  change  of  element  from  a 
point  to  a  figure  depending  upon  any  number  of  parameters, 
which  is  Pliicker's  idea  of  making  our  space  n-dimensional. 
(To  this  I  shall  yet  have  occasion  to  refer  later.) 

Another  eminent  thinker  on  the  subject  who,  in  the  main, 
holds  the  same  opinions  and  who  has  written  some  expositions  of 
the  ideas  of  Lie  and  of  his  own  views  on  the  subject,  is  the  illus- 
trious French  mathematician  Poincar6.  The  first  publication  of 
his  on  this  subject  is  the  paper  in  the  Bull,  de  la  Soc.  Math,  de 
France^  quoted  above ;  then  papers  of  his  on  the  same  subject 
appeared  in  the  Revue  Generale  des  Sciences  Pures  et  Appliquies, 
t.  Ill,  1892,  and  in  the  Revue  de  Metaphysique  et  de  Morale. 
Another  paper,  in  which  he  further  develops  and  complements 
his  views  in  the  previous  papers,  appeared  in  the  Monist,  Vol. 
IX,  1898,  translated  into  English  by  McCormack.  I  quote 
extensively  from  this  paper,  as  I  find  in  it  so  many  points  of 
agreement  with  some  of  my  own  views  —  to  which  I  have  ar- 
rived independently — about  the  relation  of  experience  and 
pure  reasoning  to  the  formation  of  our  geometrical  notions,  also 
in  relation  to  the  number  of  dimensions  of  space,  and  other 
points. 

He  begins  thus  :  "  Our  sensations  cannot  give  us  the  notion 
of  space.  That  notion  is  built  up  by  the  mind  from  elements 
preexisting  in  it,  and  external  experience  is  simply  the  occa- 
sion for  its  exercising  this  power  ....'' 

He  maintains  further  that  variations  in  our  sensations  give 
rise  to  our  notions  of  space.  We  observe  two  kinds  of  changes 
in  our  impressions,  which  we  thus  separate  into  two  classes : 

1)  External  changes,  independent  of  our  will,  and 

2)  Internal  changes,  accompanied  by  voluntary  muscular 
exertions. 

The  external  changes  again  fall  into  two  subdivisions  : 

1)  Displacements y  capable  of  being  corrected  by  an  internal 
change,  and 

2)  Altei-ationSy  or  physical  changes  not  having  this  property. 
Only  Displacements  are  the  Object  of  Geanieiry. 

An  identical  displacement  can  be  repeated  a  number  of  times. 
Hence  the  introduction  of  number. 


12 

The  ensemble  or  aggregate  of  displacements  form  a  group, 
since  the  combination  of  any  number  of  these  is  one  of  the 
aggregate. 

The  notion  of  group  could  not  be  formed  by  a  priori  reas- 
oning, but  by  experience  together  with  reasoning.  We  ah- 
stract  from  the  concrete  alterations  which  may  accompany  dis- 
placements, so  that  geometry  is  safe  from  all  revision. 

"When  experience  teaches  us  that  a  certain  phenomenon 
does  not  correspond  to  the  laws  of  the  group,  we  strike  it  from 
the  list  of  displacements.  When  it  obeys  these  laws  only  ap- 
proximately, we  consider  the  change  by  an  artificial  convention 
as  the  resultant  of  two  compound  changes.  One  is  regarded 
as  a  displacement,  rigorously  satisfying  the  laws  of  the  group, 
while  the  second  is  regarded  as  a  qualitative  alteration.  Thus 
we  say  that  solids  undergo  not  only  great  changes  of  position, 
but  also  small  thermal  alterations.'^  (Compare  with  this  Pos- 
tulate 1  and  Scholium  to  Definition  8  in  my  Dissertation.) 

"  The  fact  that  the  displacements  form  a  group  contains  in  a 
germ  a  host  of  important  consequences.  Space  must  be  homo- 
geneous ;  that  is,  all  points  are  capable  of  playing  the  same 
part  .  .  . 

"Being  homogeneous,  it  will  be  unlimited,  for  a  category 
that  is  limited  cannot  be  homogeneous,  seeing  that  the  boun- 
daries cannot  play  the  same  part  as  the  center.  But  this  does 
not  say  that  it  is  infinite,  for  a  sphere  is  an  unlimited  surface, 
and  yet  it  is  finite." 

[To  this  reasoning  I  should  object,  for  I  should  ask  :  Is  not 
a  sphere  a  bounded  body,  and  therefore  non-homogeneous  in 
the  third  dimension  ?  If  space  were  finite,  it  would  not  be 
homogeneous  in  some  dimension  ;  but  as  all  dimensions  belong 
to  space,  it  would  be  non-homogeneous  in  some  of  its  own 
dimensions ;  hence,  we  could  not  say  without  limitation  that 
any  two  displacements  form  a  new  displacement. 

In  my  proof  of  the  infinite  extent  of  the  straight  line,  I  do 
not,  however,  assume  the  infinity  of  space.  I  only  postulate 
that  "  each  point  is  capable  of  playing  the  same  part  as  any 
other,''  which  is  postulated  by  Poincar6  also,  as  being  involved 
in  the  notion  of  the  group.  Hence,  from  any  point  we  can  de- 
scribe a  sphere  with  a  distance  actually  given  by  a  previous 
construction,  if  it  is  possible  to  do  it  for  one  point  and  for  one 
given  distance.     (See  Theorem  2,  p.  67.)] 


13 

After  postulating  continuity  for  the  group  of  displacements, 
and  defining  what  is  meant  by  subgroups,  isomorphism,  inva- 
riant subgroups,  etc.,  into  which  I  cannot  go  in  detail  without 
transcribing  the  paper  as  a  whole,  Poincare  goes  on  to  say 
that  by  experience  combined  with  abstraction  we  arrive  at  the 
notion  of  a  rotative  subgroup^  or  the  ensemble  of  displacements 
which  conserve  a  certain  system  of  sensations.     Then  he  says  : 

"  By  new  experiences,  always  very  crude,  it  is  then  shown  : 

"1.  That  any  two  rotative  subgroups  have  common  dis- 
placements. 

"  2.  That  these  common  displacements,  all  interchangeable 
among  one  another,  form  a  sheaf,  which  may  be  called  a  rotative 
sheaf  (rotations  about  a  fixed  axis). 

"3.  That  any  rotative  sheaf  forms  part  not  only  of  two 
rotative  subgroups  but  of  an  infinity  of  them.  There  is  the 
origin  of  the  notion  of  the  straight  line,  as  the  rotative  sub- 
group was  the  origin  of  the  notion  of  the  point." 

[In  these  few  sentences  one  may  discover  a  somewhat  crude 
empirical  statement  of  the  facts  of  which  I  availed  myself  in 
constructing  the  straight  line  in  Theorem  2  (see  pp.  67-83).] 

He  then  goes  on  to  say  that  the  existence  of  an  invariant  stib- 
groupj  namely,  the  subgroup  of  translations,  in  which  all  dis- 
placements are  interchangeable,  is  the  only  fact  "  that  determines 
our  choice  in  favor  of  the  geometry  of  Euclid,  as  against  that  of 
Lobatchevshiy  because  the  group  that  corresponds  to  the  geometry  of 
Lobatchevski  does  not  contain  such  an  invariant  subgroup.'^ 

When  he  comes  to  the  discussion  of  dimensions,  Poincar^ 
points  out  the  distinction,  from  the  point  of  view  of  the  theory 
of  groups,  between  the  order  k  and  the  degree  n  of  a  group,  and 
states  that  the  order  k  is  the  more  important  characteristic  of  a 
group.  So  that  two  groups  can  be  isomorphic  (i.  e.,  their  opera- 
tions obey  the  same  laws  of  combination  and  hence  have  the 
same  number  of  subgroups  etc.),  and  still  be  of  different  de- 
gree, provided  their  order  k  is  the  same.  In  continuous  groups, 
in  general,  and  in  the  group  of  displacements,  in  particular,  the 
object  of  operations  ''is  the  ensemble  of  a  certain  number  n  of 
quantities  susceptible  of  being  varied  in  a  continuous  manner, 
which  quantities  are  called  coordinates."  —  "  Then,  every  infini- 
tesimal operation  of  the  group  can  be  decomposed  into  k  other 
operations  belonging  to  k  given  sheaves.  The  number  n  of  the 
coordinates  (or  of  the  dimensions)  is  then  the  degree,  and  the 


14 

number  h  of  the  components  of  an  infinitesimal  operation  is  the 
order  J  ^  —  "  The  degree  is  an  element  relatively  material  and  sec- 
ondary, and  the  order  a  formal  element/'  —  The  study  of  the 
group  is  mainly  the  study  of  its  formal  properties.  The  order 
k  corresponds  to  the  number  of  essential  parameters  of  a  group 
of  transformations.  —  "  The  group  of  displacements  is  of  the 
sixth  order."  —  The  order  k  in  case  of  R^  is  6,  since  B^  can 
have  00^  displacements.  —  As  to  the  degree  n,  it  depends  upon 
the  choice  of  the  element. 

If  we  choose  the  different  transformations  of  a  rotative  sub- 
group, we  get  a  triple  infinity  of  elements.  "  The  degree  of 
the  group  is  three.  We  have  chosen  the  point  as  the  element 
of  space  and  given  to  space  three  dimensions. 

'*  Choosing  the  different  transformations  of  a  helicoidal  sub- 
group, we  obtain  a  quadruple  infinity  of  elements.  We  have 
chosen  the  straight  line  as  the  element  of  space,  —  which  gives 
to  space  four  dimensions. 

"  Suppose,  finally,  that  we  choose  the  different  transformations 
of  a  rotative  sheaf.  The  degree  would  then  be  five.  We  have 
chosen  as  the  element  of  space  the  figure  formed  by  a  straight 
line  and  a  point  on  that  straight  line.  Space  would  have  five 
dimensions. 

"The  introduction  of  a  group  more  or  less  complicated,  ap- 
pears to  be  absolutely  necessary.  Every  purely  statical  theory 
of  the  number  of  dimensions  will  give  rise  to  many  difficulties, 
and  it  will  always  be  necessary  to  fall  back  upon  a  dynamical 
theory." 

[I  wish  to  observe  here  that  the  deduction  of  the  number  of 
dimensions  in  my  Dissertation  is  based  upon  kinematical  prin- 
ciples.] 

"  When  I  pronounce  the  word  ^  length/  a  word  which  we 
frequently  do  not  think  necessary  to  define,  I  implicitly  assume 
that  the  figure  formed  by  two  points  is  not  always  superposable 
upon  that  which  is  formed  by  two  other  points ;  for,  otherwise, 
any  two  lengths  whatever  would  be  equal  to  each  other.  Now, 
this  is  an  important  property  of  our  group. 

"I  implicitly  enunciate  a  similar  hypothesis  when  I  pronounce 
the  word  *  angle.' " 

I  have  still  to  quote  his  ideas  concerning  contradictions  in 
geometry,  as  I  think  they  are  of  cardinal  importance.  Here  is 
what  he  says  : 


15 

"  In  following  up  all  the  consequences  of  the  different 
geometrical  axioms  are  we  never  led  to  contradictions  ?  .  .  . 
The  axioms  are  conventions.  Is  it  certain  that  all  these  con- 
ventions are  compatible? 

"  These  conventions,  it  is  true,  have  all  been  suggested  to  us 
by  experience,  but  by  crude  experience.  We  discover  that 
certain  laws  are  approximately  verified,  and  we  decompose  the 
observed  phenomenon  conventionally  into  two  others  :  a  purely 
geometrical  phenomenon,  which  exactly  obeys  these  laws ;  and 
a  very  minute  disturbing  phenomenon. 

"Is  it  certain  that  this  decomposition  is  always  permissible ? 
It  is  certain  that  these  laws  are  approximately  compatible,  for 
experience  shows  that  they  are  all  approximately  realized  at 
one  and  the  same  time  in  nature.  But  is  it  certain  that  they 
would  be  compatible  if  they  were  absolutely  rigorous  ? 

"  Fo7'  us  the  question  is  no  longer  doubtful.  Analytical  geom- 
etry has  been  securely  established,  and  all  the  axioms  have  been 
introduced  into  the  equations  which  serve  u^  as  its  point  of  depar- 
ture; we  could  not  have  written  these  equations  if  the  axioms  Jiad 
been  contradictory.  Now  that  the  equations  are  written,  they  can 
be  combined  in  all  possible  manners ;  analysis  is  the  guarantee 
that  contradictions  shall  not  be  introduced.'^ 


We  see  thus  that  both  Lie  and  Poincar^  are  of  the  opinion, 
that  the  question  about  the  foundations  of  geometry  represents 
a  more  concrete  and,  therefore,  more  easily  manageable  prob- 
lem than  Klein  and  Killing  and  some  others  are  willing  to 
grant.  Neither  of  the  former  mathematicians  allows  an  infin- 
ity of  contradictory  geometries,  and  Poincar^  even  gives  some 
reasons  for  our  choice  in  favor  of  the  Euclidian  geometry.  He 
thinks  only  that  this  is  solely  due  to  the  mode  of  experience 
we  have  of  space,  and  when  he  speaks  of  hypothetical  beings, 
whose  experience  might  have  led  them  to  a  predilection  for  the 
geometry  of  Lobatchevski,  he  certainly  is  right,  in  the  sense 
that  our  geometrical  notions,  siich  as  they  are,  are  not  altogether 
independent  of  the  mode  of  experience  we  have,  and  the  nature 
of  the  universe  we  live  in.     In  Poincar^'s  own  words : 

"  It  is  our  mind  that  furnishes  a  category  for  nature.  But 
this  category  is  not  a  bed  of  Procrustes  into  which  we  viole^Uly 
force  nature,  mutilating  her  as  our  needs  require.     We  offer  to 


16 

nature  a  choice  of  beds,  among  which  we  choose  the  couch  best 
suited  to  her  stature/' 


Now,  this  view  of  the  matter  is  certainly  more  encouraging 
to  the  one  who  would  venture  to  find,  by  methods  more  ele- 
mentary and,  consequently,  more  legitimate  for  the  given  pur- 
pose than  those  based  upon  the  laws  of  continuous  groups,  ex- 
actly which  couch  is  the  most  suitable  for  nature's  stature,  and 
even  under  what  conditions  the  other  couches  may  become  just 
as  suitable. 

Let  us  now  turn  to  Riemann,  whose  paper  on  the  founda- 
tions of  geometry  seems  to  have  been  the  occasion  (apparently 
unintended  by  the  author),*  of  many  a  misconception  sanc- 
tioned by  his  name.  Riemann  himself,  as  well  as  can  be  gath- 
ered from  the  fact  that  he  tried  to  find  the  laws  of  free  mobility 
in  manifolds  of  n  dimensions,  considering  space  as  a  special 
CQjSe  of  such  manifolds,  where  n=:  S,  seems  to  have  been  of  the 
opinion  that  geometry,  as  a  science  of  space  and  spacial  magni- 
tudes alone,  must  be  one  and  only  one,  although  he  did  not  de- 
cide in  favor  of  any  of  the  three  possible  systems.  He  thought, 
at  any  rate,  that  the  discovery  of  the  truth  concerning  the  natwe 
of  the  geometry  of  our  space  repr^esents  a  concrete  and  not  unsolv- 
ahle  scientific  p)rohlem.  He  stated  expressly  that  the  solution  of 
this  problem  was  not  to  be  found  in  investigations  of  such  a 
general  character  as  was  his  own  about  manifolds  in  general  — 
a  thing  which  his  followers  have  not  always  heeded  sufficiently 
—  since  space  was  for  him  a  manifold  of  special  character, 
whose  science  alone  he  called  geometry,  as  distinguished  from 
the  science  of  the  general  laws  of  manifolds,  which,  accord- 
ing to  him,  belongs  to  analysis  and  the  theory  of  functions. 
So  he  says  on  p.  258  of  his  "  Math.  Werke  "  :  "  These  magni- 
tudinal  relations  "  (of  multiply-extended  manifolds  in  general) 
"  admit  of  investigation  only  in  terms  of  abstract  quantity,  their 
natural  connection  being  representable  by  formulce  ;  under  certain 
assumptions  they  can,  however,  be  decomposed  into  relations, 
which,  taken  separately,  are  capable  of  geometrical  represeittation, 
and  through  this  it  becomes  possible  to  express  geometrically  the 

*  See  above,  note  to  p.  4. 


17 

remit  of  the,  calculation.  So  that,  although  an  abstract  investi- 
gation, by  means  of  formulae,  remains  unavoidable,  it  will  still 
be  possible  to  invest  its  final  results  in  geometrical  attire." 
And  on  page  256  he  says,  "  More  frequent  occasions  for  cre- 
ating and  developing  these  conceptions  (of  multiply-extended 
manifolds)  we  find  only  in  the  higher  mathematics." 

His  own  investigation  he  considered  useful  only  in  so  much 
as  it  threw  light  upon  the  extent  and  nature  of  the  implicit  a«- 
sumptions  of  geometry  and  upon  the  many  questions  of  measure- 
ment in  the  wider  region  of  multiply-extended  manifolds,  upon 
which  these  assumptions  touch,  and  which,  apparently,  have 
escaped  the  attention  of  his  predecessors.  He  says  on  p.  268  : 
"  Such  investigations  which,  like  the  one  here  carried  through, 
start  from  general  conceptions,  can  serve  only  the  end  that 
this  work  (the  investigation  of  the  real  facts  underlying  our 
notions  of  space  and  of  its  magnitudinal  relations)  shall  not  be 
hampered  by  too  narrow  conceptions,  and  that  the  progress  of 
discovery  of  the  connection  of  things  should  not  be  impeded  by 
the  burden  of  inherited  prejudice." — He  looked,  however,  for 
the  solution  of  this  problem  in  the  wrong  direction,  when  he 
thought  that  some  physical  hypothesis  which  may  in  time  prove 
necessary,  to  account  for  certain,  as  yet  unexplained,  physical 
phenomena  in  the  realm  of  the  infinitely  small,  might  also 
throw  some  light  upon  the  true  nature  of  geometry.  He  cer- 
tainly erred  in  respect  of  this  physical  hypothesis  of  the  geo- 
metrical properties  of  our  space.*  They  could  lead  to  no  better 
results  than  astronomical  observations  upon  the  stellar  paral- 
laxes, instituted  with  the  purpose  of  finding  some  testimony  in 
the  immensely  large  triangles  concerning  the  amount  by  which 
the  sum  of  the  three  angles  of  a  triangle  is  less  than  two  right 
angles, — as  is  very  evident  from  the  truly  philosophical  treat- 
ment of  this  subject  by  Poincar^.  But  be  this  as  it  may,  the 
fact  still  remains  that  Riemann,  in  the  first  place,  regarded 
space  as  an  unbounded  manifold  of  three  dimensions,  and  spoke 
of  it  as  being  an  empirical  certainty  greater  than  any  other  we 
have,  and,  secondly,  thought  that  the  problem  as  to  the  admissi- 
bility of  the  propositions  of  the  Euclidian  geometry  beyond  the 

*'*It  must  be,  therefore,  either  that  the  realities  which  lie  at  the  basis 
of  space  form  a  discrete  manifold,  or  that  the  fonndation  of  its  magnitudinal 
relations  ought  to  be  looked  for  outside,  in  the  binding  forces  working  upon 
it."  (p.  268). 


18 

bounds  of  observation  was  still  an  unsolved,  but  not  unsolvahlcy 

scientifixi  problem. 

Some  of  the  results  arrived  at  by  Riemann  are  : 

1.  The  simplest  form  of  the  linear  element  in  any  n-fold- 

extended  manifold  admitting  of  measurement,  is 


ds  =  V^OL^jdx.dXj^, 

where  the  a's  are  continuous  functions  of  the  ic's ;  of  these,  n  func- 
tions can  be  taken  arbitrarily,  and  n{n  —  l)/2  are  fixed  by  the 
nature  of  the  manifold.  In  space,  for  instance,  even  if  it  were 
curved,  three  of  the  a's  could  be  taken  =  0,  each,  and  the  rest  would 
have  to  take  their  chances,  which  would  depend  upon  the  nature 
of  the  curvature.  At  this  stage  of  his  investigation,  Riemann 
seems  to  assume  that  space  and  the  plane  are  flat  manifolds,  so 
that  their  linear  elements  can  be  brought  to  the  form  of  V^dx^ 
(see  p.  200,  Werke).  It  would  seem,  therefore,  that  when  later 
he  speaks  of  the  possibility  of  space-curvature,  and  of  a  physical 
investigation  in  the  realm  of  the  infinitely  small,  he  means 
rather  that,  since  he  does  not  see  any  logical,  a  'priori  necessity 
of  the  necessary  and  sufficient  assumptions  of  the  Euclidian 
geometry,  which  he  establishes  iu  §  1  of  Art.  Ill,  p.  205,  he 
hopes  to  find  an  explanation  of  their  necessity  in  physics,  as 
he  does  not  hope  to  find  light  on  this  subject  in  geometry 
proper,  her  realm  being  only  the  finite. 

2.  In  an  n-manifold  we  can  construct  at  each  point  oc"~^ 
geodesies ;  then  a  surface-element  is  determined  by  any  two  of 
these  given  by  their  linear  elements,  when  these  are  prolonged 
until  they  become  finite  geodesies.  In  other  words,  as  Klein 
puts  it  in  his  "  Nicht-Euklid.  Geom.,^'  p.  211,  we  have  to  con 
sider  the  collectivity  of  geodesies  whose  linear  elements 

ds.=  \'d's.^  \"d"s., 

or  such  whose  initial  directions  are  in  the  same  linear  manifold 
with  the  two  given  ones.  Each  of  the  surfaces  thus  obtained 
will  have  its  own  initial  Gaussian  curvature,  which  Riemann 
defines  as  the  curvature  of  the  n-manifold  at  the  given  point  in 
the  given  surface-direction. 

3.  A  manifold  of  constant  curvature  is  such  as  has  the  Gauss- 
ian curvature  in  its  surface-element  the  same  at  all  points  and 
in  all  surface-directions.     But  the  nature  of  the  manifold  at  a 


19 

point  will  be  completely  determined  as  soon  as  the  surface-cur- 
vature is  given  in  7i(n  —  l)/2  surface-directions. 

4.  Only  a  manifold  of  constant  curvature  allows  free  mobility 
of  figures,  and  if  the  Gaussian  curvature  be  denoted  by  ot,  the 
linear  element  of  such  a  manifold  can  be  reduced  to  the  form  of 


V^dx'. 


5.  All  metrical  relations  of  the  manifold  depend  upon  the 
value  of  the  curvature.  The  number  of  ways  in  which  an 
n-manifold  can  move  in  itself  without  deformation  is 

n{n  H-  1)  _    2  _  K^  ~  ^)  _ 


number  of  coordinates  minus  number  of  distances  between  n 
points. 

Rieraann  then  gives  three  possible  forms  of  the  conditions 
necessary  and  sufficient  to  determine  the  measure-relations  of 
space,  as  distinguished  from  all  other  three-dimensional  mani- 
folds admitting  of  measurements  and  flat  in  their  smallest 
parts,  i.  e.y  such  whose  line-length  is  independent  of  position 
and  in  which  the  linear  element  is  expressible  as  the  square 
root  of  a  positive  differential  expression  of  the  second  degree. 

1°.  The  Gaussian  curvature  in  three  surface-directions  is  zero 
at  each  point ;  or,  otherwise,  the  metric  relations  of  space  are 
completely  determined,  if  the  sum  of  the  three  angles  of  a  tri- 
angle is  always  equal  to  two  right  angles. 

2°.  Besides  the  independence  of  line-length  from  position,  we 
may  assume  with  Euclid  the  existence  of  rigid  bodies,  inde- 
pendent of  position, — which  is  equivalent  to  postulating  constant 
curvature.  The  sum  of  the  three  angles  in  all  triangles  is  then 
determined,  when  it  is  known  in  one  triangle. 

3°.  We  may  assume  not  only  the  independence  of  line- 
length  from  position,  but  also  the  independence  of  length  and 
direction  of  lines  from  position. 

Each  of  these  three  alternatives  adds  something  to  the  prop- 
erties of  a  manifold,  flat  in  its  smallest  portions.  The  first 
and  last  lead  at  once  to  the  Euclidian  geometry ;  the  middle 
one  allows  the  possibility  of  all   three   different  geometries, 


20 

according  as  the  sum  of  the  angles  of  a  triangle  is  greater 
than,  equal  to,  or  less  than,  two  right  angles.* 

Now,  then,  the  position  and  the  opinions  of  this  second  cate- 
gory of  mathematicians  (E-iemann,  Lie,  and  Poincare),  and 
especially  those  of  Lie,  seem  to  indicate  that  the  task  of  estab- 
lishing an  efficient  system  of  axioms  is  not  perfectly  hopeless. 
And  if  the  reader  admit  with  Lie  that  but  few  investigations 
,have  materially  furthered  the  problem  concerning  the  founda- 
tions of  geometry,  he  may  find  it  of  interest  to  read  through 
the  present  memoir,  even  if  its  purpose  is  disclosed  at  the  very 
beginning  to  be  —  the  establishment  of  such  a  system  and  the 
restoration  of  a  goodly  part  of  the  old  prestige  to  the  origin 
and  foundations  of  the  geometrical  science. 


It  may  not  be  amiss  here,  in  this  connection,  to  remind  the 
generously  disposed  and  impartial  reader  that  this  problem, 
besides  its  philosophic  interest,  is  also  of  importance  from  a 
purely  mathematical  point  of  view.  The  fact  which  will  sub- 
stantiate my  statement  is,  indeed,  very  well  known  to  mathe- 
maticians who  have  familiarized  themselves,  at  least  superficially, 
with  the  non-Euclidian  geometry,  although  little  stress  is  put 
upon  its  bearings  by  those  mathematical  writers  on  the  subject, 
who,  having  satisfied  themselves  that  Avithin  the  bounds  of  our 
limited  experience  the  Euclidian  geometry  holds,  concluded 
that  beyond  these  limits  actual  deviations  of  the  metrical  rela- 
tions of  space  may  take  place,  of  which  we  are  not  bound  to 
take  heed  in  our  analytical  geometry,  as  long  as  we  intend  to 
avail  ourselves  of  its  results  in  actual  practice  only.  Already 
the  earliest  non-Euclidians,  and  among  them  the  two  great 
founders  of  the  hyperbolic  geometry,  Lobatchevski  and  Bolyai, 
have  made  it  clear  that  the  theory  of  proportion  and  similar 
figures  is  based  upon  the  parabolic  system  of  measurement,  and 
that  it  has  no  meaning  when  the  Euclidian  postulate  of  paral- 
lels does  not  hold.  In  fact,  they  have  both  given  formulae  for 
the  solution  of  rectilinear  triangles,  perfectly  analogous  to  those 
of  the  spherical  trigonometry,  f     Further,  Bolyai  has  shown 

*See  Lie,  "Transformationsgruppen,"  Vol.  Ill,  p.  497,  where  he  finds 
this  paragraph  in  Kiemann's  paper  (§  1  of  art.  Ill)  not  clear. 

fSee  Lobatchevski,  "Theory  of  Parallels,"  translated  by  Halstead,  pp. 
35-45.  Also,  "  Urkunden  zur  Geschichte  der  Nioht-Euklid.  Geom."  F. 
Engel,  pp.  216-235. 


21 

that,  by  assuming  the  hyperbolic  geometry  to  be  true,  the  prob- 
lem of  the  squaring  of  the  circle  presents  no  difficulty.*  It  is, 
therefore,  evident  that  the  establishment  of  the  Euclidian 
geometry  on  a  basis  more  rational  than  mere  empiricism  even 
if  very  accurate,  still  remains  a  desideratum. 

It  will,  however,  become  incumbent  upon  me  to  explain  my 
own  point  of  view  in  this  matter,  and  give  in  outline  the  re- 
sults at  which  I  have  arrived,  and  also  to  throw  some  light 
upon  the  methods  pursued  in  this  dissertation,  as  well  as  to 
present  the  reasons  which,  according  to  the  best  of  my  judg- 
ment, can  be  assigned  to  the  final  success  with  which  these 
methods  have  been  rewarded. 

To  give  my  own  views  upon  the  r6le  of  experience  and  reason 
in  the  formation  of  our  geometrical  conceptions  would,  I  think, 
only  be  a  repetition  of  what  is  stated  more  or  less  explicitly  in 
my  introductory  chapter  on  dimensions,  as  well  as  a  repetition 
of  many  excellent  remarks  of  Poincar^  in  his  paper  in  the 
Monist,  which  I  have  allowed  myself  to  quote  so  extensively. 
In  a  few  words  these  views  may,  however,  be  summarized 
thus : — 

As  in  all  pure  sciences,  our  fundamental  conceptions  in 
geometry  are  formed  by  experience  helped  on  by  pure  reasoning^ 
which  abstracts  from  certain  unessential  irregularities  in  the  rough 
data  of  experience,  by  reducing  certain  general  norms  to  ideal 
forms,  not  admitting  of  exception.  The  exceptions,  indeed,  are 
purposely  eliminated  by  ascribing  to  them  some  other  causes, 
which  are  not  the  subject  of  the  given  investigation.  So,  for 
instance,  in  mechanics,  the  fact  that  no  body  in  actual  experi- 
ence, possessing  a  certain  momentum,  can  go  on  and  move  for- 
ever, does  not  bother  the  physicist,  who  postulates  the  first  law 
of  Newton,  and  ascribes  the  stopping  of  the  body  or  the  retar- 
dation of  its  motion  to  external  causes,  like  frictional  resistance, 
etc.  Similarly,  if  ideal  solids  are  postulated  in  geometry,  the 
deformation  which  natural  solids  undergo  in  motion  is  ascribed 
to  physical  causes,  and  not  to  properties  of  space.! 

Further,  I  think  that  we  cannot  start  simply  with  an  axiom 
—  that  space  is  a  number-manifold,  i.  e.,  each  point  in  it  can 

*See  "  Science  Absolute  of  Space  "  by  John  Bolyai,  Halstead's  transla- 
tion, p.  47. 

t  See  postulate  1,  Definition  6,  and  Scholium  to  Definition  8,  of  the  intro- 
ductory chapter  of  my  Dissertation  (pp.  40,  41). 


22 

be  determined  by  three  coordinates,  or  three  numbers  which 
can  be  made  to  vary  continuously.  I  think  rather  that  space 
ought  to  be  proved  capable  of  being  made  a  number-manifold, 
and  the  best  starting  point  in  this  direction  is,  according  to  my 
opinion,  to  be  found  in  the  simple  physical  facts  of  impenetra- 
bility, rigidity,  and  divisibility  of  bodies,  each  in  geometry  be- 
ing idealized.  The  bulk  of  a  given  portion  of  space,  bounded 
on  all  sides,  can  certainly  be  represented  by  a  number,  showing 
how  many  times  it  will  contain  a  smaller  bulk  of  definite  shape. 
By  considering  the  smaller  bulk  as  rigid,  or  such  in  which 
internal  motion  or  rearrangement  of  parts  is  excluded,  and 
making  this  smaller  bulk  take  up  all  possible  positions  within 
the  larger  bounded  portion  of  space,  we  observe  that  no  matter 
whence  it  he  placed  within  the  larger  one,  it  always  occupies  or 
Jills  up  the  same  numerical  portion  of  the  larger  bulk.  The  num- 
ber of  other  bulks  like  the  smaller,  necessary  to  fill  up  the 
larger  completely,  besides  the  smaller  one  itself,  or  the  num- 
ber of  places  the  smaller  can  be  made  to  occupy  within  the 
larger,  such  that  no  two  have  any  portion  in  common,  is  always 
the  same.  And  this  is  true  also  when  the  smaller  bulk  is 
broken  up  into  infinitely  small  portions,  free  to  change  position 
with  respect  to  one  another,  but  still  capable  of  filling  up  com- 
pletely the  same  space ;  or,  in  other  words,  when  the  smaller 
bulk  is  allowed  to  change  its  form  in  all  possible  ways,  so  as  to 
retain  only  impenetrability.  Equal  bulks  are  then  measured 
by  equal  spaces  of  same  shape,  which  they  are  capable  of  filling.* 
We  postulate  that  this  be  true  for  any  bounded  space  and 
for  any  small  bulk  which  is  placed  in  the  larger  one,  in  any 
position,  —  that  there  should  always  be  the  same  numerical  re- 
lation between  the  smaller  and  the  larger  bulk  as  soon  as  these 
are  given,  as  a  rigid  solid,  on  the  one  hand,  and  a  bounded 
vacuum  in  which  the  first  is  to  lie  in  any  position,  on  the  other 
hand.  We  arrive  at  the  notion  of  congruent  portions  within 
the  bounded  space,  meaning  such  which  the  same  solid  fills  up 
to  the  exclusion  of  others,  —  and  by  considering,  besides,  very 
small  portions  of  the  smaller  bulk,  their  number  and  disposi- 
tion with  respect  to  one  another  are  seen  not  to  change  as  long 
as  rigidity  of  form  is  postulated  for  the  whole.     So  that,  after 

*  For  a  complete  and  rigorous  treatment  of  this  question,  the  reader  is 
referred  to  the  introductory  chapter,  Scholium  to  Definition  8,  pp.  41-44. 
Here  is  possible  only  a  short  indication  of  the  procedure. 


23 

we  have  worked  ourselves  up  to  the  notion  of  surface,  line,  and 
point,  as  done  in  the  introductory  chapter,  rigidity  is  seen  to 
imply  that  no  two  points,  separated  in  one  position  of  a  rigid 
body,  become  ever  coincident  on  account  of  change  of  position 
of  the  solid.  And,  moreover,  the  same  continuous  series  of 
separated  points  of  the  solid  must  be  capable  of  being  con- 
structed between  any  two  given  points  of  the  solid  in  any  one 
of  its  positions  as  in  any  other. 

But  this  is  only  a  starting  point.  We  must  further  make 
clear  to  ourselves  what  we  understand  when  we  say  that  space 
is  a  three-dimensional  manifold,  considering  a  point  as  its  ele- 
ment, and  whether  there  is  any  sense  in  looking  for  a  fourth 
dimension,  not  directly  given  by  experience.  It  will  appear 
from  the  treatment  of  the  question  in  the  introductory  chapter 
that  the  tridimensionality  of  space  is  actually  postulated  by  the 
definition  of  a  point,  and  that,  therefore,  to  look  for  a  fourth 
dimension,  without  changing  its  element  from  that  which  lies  at 
the  basis  of  the  metrical  geometry  of  Euclid  to  some  other 
geometrical  object  (which  is,  in  fact,  a  figure  in  the  Euclidian 
sense,  depending  upon  a  certain  number  of  parameters),  as  is 
done,  for  instance,  in  Pliicker's  line-geometry,  —  is  a  contradic- 
tion in  terms.  Finally,  from  the  same  principle  of  rigidity, 
the  notion  of  distance  as  an  invariable  relation  between  two  points 
in  rigid  connection  or  in  fixed  space,  is  easily  derived  by  a  defi- 
nition which  makes  use  of  the  principle  of  superposition. 

Next,  continuity  must  be  postulated,*  and  then  we  must  show 
how  distances  can  be  added  and  subtracted,  and  whether  there 
is  a  line,  or  a  one-dimensional  manifold,  in  space,  capable  of  rep- 
resenting by  the  actual  distances  of  its  points  all  possible  dis- 
tances arrived  at  by  addition  and  subtraction,  and  whether  this 
can  be  done  in  a  unique  way.  It  appears,  that  from  this  property 
alone  a  notion  of  the  straight  line  can  be  deduced,  which  will 
have  all  other  properties  of  the  straight  line,  commonly  postu- 
lated for  it  in  the  Euclidian  geometry ;  the  construction,  more- 
over, based  upon  this  property,  will  make  it  a  number-manifold  f 
of  infinite  extent,  such  as  can  in  no  way  be  mixed  up  with  a 
geodesic  returning  into  itself  at  a  finite  distance.     This,  in  a 

*  See  axiom  1,  p.  63. 

fThe  construction  makes  the  straight  line  a  number-manifold  in  the 
Cantor-sense,  since,  as  we  can  construct  all  possible  sums  of  all  possible  ra- 
tional numbers,  we  can  construct  the  irrational  numbers  by  sequences,  in  the 
way  it  is  done  by  Cantor  for  pure  numbers.     Math.  Ann.f  t.  5,  pp.  123-128. 


24 

certain  way,  disposes  of  the  so-called  elliptic  geometry  of  the 
straight  line.  Further,  the  line  so  constructed  will  prove  to 
be  both  an  axis  of  rotation  and  an  axis  of  helicoidal  motion,  or, 
in  Poincar6  and  Lie's  language,  admitting  either  the  transfor- 
mations of  a  rotative  sheaf,  or  those  of  a  helicoidal  subgroup,  ac- 
cording as  one  point,  at  least,  upon  the  line  is  taken  as  an  in- 
variant point,  or  none,  at  a  finite  distance,  is  taken  as  an  inva- 
riant point  —  the  line  being  able  to  slide  upon  itself,  while 
all  points  in  rigid  connection  with  the  line,  but  outside  it, 
being  free  only  to  twist,  i.  e.,  to  move  with  a  screw-like  motion. 
The  notion  of  distance,  as  thus  defined,  is,  of  course,  of  a 
very  abstract  nature,  and  does  not  depend  upon  the  paths 
which  either  of  two  non-coincident  points  can  be  made  to  pass 
in  a  given  surface  from  its  own  position  to  the  position  of 
the  second,  nor  upon  any  given  position  of  the  point-couple  in 
space,  but  only  upon  the  relative  position  of  the  two  points 
themselves.  It  is  simply  a  fact  of  experience  that  a  pair  of 
points  in  a  solid  are  capable  of  coincidence  only  mth  certain  de- 
terminate couples  of  points  in  other  solids  or  vacant  space,  and 
this  fact  of  congruence  or  non-congruence  is  the  only  factor 
determining  equality  or  non-equality  of  distances.  It  is  only 
after  it  has  been  proved  that  this  geometrical  magnitude  is 
representable  uniquely  and  perfectly  by  some  line  (a  priori,  a 
surface  or  a  volume  might  perhaps  have  been  found  more 
capable  to  represent  this  magnitude,  as  happens,  for  instance, 
with  the  angular  magnitude,  which  is  equally  well  represented 
by  a  portion  of  a  circular  arc  as  by  the  area  of  a  sector  of  the 
circle  of  radius  unity,  and  as,  for  instance,  the  solid  angle, 
considered  as  a  geometrical  magnitude,  may  be  measured  equally 
by  the  area  of  a  spherical  surface  of  radius  unity  whose  center 
is  the  vertex  of  the  angle,  or  by  the  corresponding  spherical 
sector ;  so  that  it  is  only  an  accident,  having,  of  course,  its  rea- 
sons in  the  nature  of  things  a  posteriori,  that  distance  as  a 
geometrical  magnitude  has  for  its  representation  a  line),  it  is 
only  after  this  fact  has  been  established,  that  any  curve  can  be 
broken  up  into  linear  elements  ds,  each  of  which  is  comparable 
with  the  elements  of  three  given  straight  lines  dx,  dy,  dz,  and 
can  be  expressed  in  terms  of  these.  So  that  any  complicated 
expression  for  a  linear  element  of  some  curve  in  space,  in  terms 
of  dx,  dy,  dzy  say,  ds  —f(dx,  dy,  dz),  must  necessarily  be  based, 
in  the  first  instance,  upon  a  certain  relation,  which  could  be 


25 

considered  the  simplest  and  which  would  be  formed  exactly  in 
the  same  way  as  a  finite  distance  is  expressed  in  terms  of  three 
other  finite  distances,  which  must  be  taken  as  parameters  in  the 
case  of  tridimensional  space. 

I  wish  here  to  call  attention  to  the  fact  that  the  deduction  of 
the  existence  of  the  straight  line  in  space,  not  as  a  line  in  the 
plane,  as  far  as  I  am  aware,  seems  never  to  have  constituted  a 
serious  problem  with  mathematicians.  This  is,  perhaps,  the 
only  reason  why  the  straight  line  has  always  been  regarded  as 
a  geodesic  which  is  determined  by  two  points  in  the  surface  to 
which  it  belongs,  i.  e.,  as  a  geodesic  in  some  plane.  The  plane 
is  postulated  or  constructed  before  the  straight  line,  and  angles 
and  triangles  and  circles  are  regarded  not  only  as  plane  figures, 
that  iSy  such  as  can  lie  in  a  plane,  but  also  as  figures  constructed 
in  a  plane.  The  distinction,  according  to  my  mind,  is  not  at 
all  trivial,  since  the  existence  of  a  plane  can  be  proved  with 
perfect  rigor  only  after  a  number  of  theorems  concerning 
angles,  triangles,  and  circles,  have  been  established  for  these 
figures  in  space.  Then  only,  according  to  my  opinion,  we 
ought  to  prove  that  these  simple  figures,  as  well  as  the  straight 
line,  are  plane  figures. 

The  plane,  as  constructed  from  a  certain  origin,  must  then  he 
shown  to  he  capahle  of  moving  upon  itself  in  a  triply-infinite  num- 
ber of  ways,  and  also  of  coincidence  with  itself  when  its  two  sides 
are  interchanged.  The  first  will  establish  the  legitimacy  of 
what  is  called  in  analytic  geometry  change  of  origin  and  change 
of  axes  ;  the  second,  a  revolution  of  the  plane  through  an  angle 
TT,  which  appears  in  the  theory  of  groups  to  need  special  sub- 
groups of  displacements.  (See  Lie,  "  Continuierliche  Gruppen,^' 
p.  101.  The  first  kind  of  displacements  form  the  group  of  con- 
gruent figures,  the  second,  the  group  of  symmetrical  figures.*) 
In  the  Euclidian  geometry  both  processes  are  invariably  used 
in  superposition  of  figures  for  demonstrations.  In  spherical 
and  pseudospherical  geometry  this  is  also  practised,  with  the 
understanding  that  bending  without  stretching  is  postulated. 
In  spherical  geometry  bending  is  necessary  only  for  making  the 
inner  side  of  a  portion  of  a  spherical  surface  coincide  with  the 
outer  side  of  the  same  surface,  or  for  the  purpose  of  applying 

*  Group  of  congruent  figures, a?i  =  a;  cos  a  —  y  sin  a  +  a,  yi  =  «  sin  a  -{- 

y  cos  a  +  6  ;  group  of  symmetrical  figures, a^i  =^  a  cos  a  +  y  sin  a  +  «i  yi  = 

X  sin  a  —  y  cos  (x-\-h. 


26 

its  figures  to  figures  fi^rmed  upon  other  surfaces  of  equal  con- 
stant curvature ;  in  case  of  pseudospherical  geometry,  i,  e.,  sur- 
faces of  constant  negative  curvature,  no  superposition  of  parts 
of  different  regions,  even  on  the  same  side  of  the  same  surface, 
would  be  possible  without  bending.  And  it  was  just  by  ab- 
stracting from  rigidity,  in  so  much  as  bending  without  stretch- 
ing was  allowed,  that  Beltrami,  in  his  "Saggio^^*  and  in  his 
"  Teoria  fondamentale,''  was  able  to  prove  that  Lobatchevski's 
geometry  holds  good,  in  our  Euclidian  space,  upon  surfaces  of 
constant  negative  curvature,  which  he  first  named  pseudo- 
spherical  surfaces. — "  The  fundamental  criterion  of  the  demon- 
strations in  the  elementary  (Euclidian)  geometry,''  Beltrami 
begins  his  investigation  in  his  "  Saggio,''  "  consists  in  superpo- 
sition of  figures.  The  criterion  is  applicable  not  only  to  the 
plane,  but  also  to  all  surfaces  upon  which  there  can  exist  equal 
figures  in  different  positions,  that  is  to  say,  to  all  surfaces  whose 
any  portion  can  by  means  of  simple  flexion  be  applied  to  any 
other  portion  of  the  same  surface.  We  see,  in  fact,  that  the 
rigidity  of  the  surfaces  upon  which  the  figures  are  traced,  is 
not  an  essential  condition  for  the  application  of  this  criterion ; 
for  instance,  the  exactitude  of  the  plane  Euclidian  geometry 
would  not  become  deteriorated,  if  we  should  begin  by  conceiv- 
ing the  figures  traced  upon  the  surface  of  a  cylinder  or  a  cone, 
instead  of  a  plane." 

Stating  then  that  the  surfaces  whose  figures  have  a  structure 
independent  of  position,  and  hence  allowing  the  principle  of 
superposition  without  restriction,  are  those  of  constant  curvature 
only,  he  goes  on  to  say  : 

"  The  most  important  element  of  figures  is  the  straight  line. 
The  specific  characteristic  of  this  line  is  that  it  is  completely 
determined  by  two  of  its  points,  so  that  two  straight  lines  can- 
not pass  through  two  points  in  space  without  their  coinciding 
in  all  their  extent.  In  'plane  geometry ,  however y  this  principle  is 
used  only  in  the  following  form : 

"  In  making  coincide  two  planes,  in  each  of  which  there  is  a 
straight  line,  it  is  sufficient  that  the  two  lines  coincide  in  two  points, 
in  order  that  they  coincide  in  the  whole  of  their  extent, 

"  Now,  this  propefi^ty  the  plane  has  in  common  with  all  surfaces 
of  constant  curvature,  whetx,  instead  of  the  straight  lines,  we  take 

***Saggiodi  interpretazione  della  geometria  non-Euolidea, "  Oiomale  di 
Mathematiche,  1868,  t.  VI  (see  note  above,  p.  4). 


27 

the  geodesies,  ,  .  ,' If  we  make  coincide  two  surfaces  of  constant 
and  equal  curvature,  so  that  two  of  their  geodesic  lines  have  two 
points  in  common,  these  lines  will  coincide  in  all  their  extent, 

"  It  follows  that,  excluding  the  cases  where  this  property  is  sub- 
ject to  exceptions,  the  theorems  of  planimetry  whieh  are  proved  by 
the  principle  of  superposition  and  the  postulate  of  the  straight  line 
for  plane  figures,  are  true  also  for  figures  formed  in  an  analogous 
way,  upon  surfaces  of  constant  curvature,  by  geodesic  lines, 

"  TJpmi  this  are  based  the  many  analogies  between  the  geometry 
on  a  plane  and  that  on  a  sphere,  the  straight  lines  corresponding 
to  geodesies,  i.  e.,  to  arcs  of  great  circles.  For  a  sphere,  haW' 
ever,  there  exist  exceptions,  for  any  two  poi7its  diametrically  op' 
posite,  or  antipodal  points,  do  not  determine  a  geodesic  without 
ambiguity,  since  through  such  points  an  infinity  of  great  circles 
will  pass.  This  is  a  reason  why  certain  theorefins  in  plane  geom- 
etry are  not  true  for  the  sphere,  as,  for  instance,  the  theorem  that 
two  perpendiculars  to  the  same  line  do  not  meet,'' 

Beltrami  further  makes  clear  that  the  basis  of  investigation  in 
plane  geometry  is  too  general,  if,  as  usually  done,  the  only  facts 
lying  at  this  basis  are  taken  to  be  the  principle  of  superposition 
and  the  postulate  of  the  straight  line.  The  results  of  the  demon- 
strations  must  exist  whenever  this  principle  and  this  postulate,  are 
true.  They  must,  evidently,  be  true  for  surfaces  of  constant  cur- 
vature, in  which  the  postulate  of  the  straight  line  holds  without 
exception. 

Now,  the  purpose  of  Beltrami's  investigation  was  precisely  to 
show  that  this  postulate  does  not  admit  of  exceptions  in  case  of 
surfoMes  of  constant  negative  curvature.  And,  in  his  own  words, — 
"  If  we  can  prove  that  such  exceptions  do  not  exist  for  these  sur- 
faces, it  becomes  evident  that  the  theorems  of  the  non-Euclidian 
planimetry  hold  without  restriction  upon  such  surfaces.  And  then 
certain  results  which  seem  incompatible  with  the  hypothesis  of  a 
plane,  may  become  corweivable  upon  such  a  surface  and  obtain 
ther-eby  an  explanation,  not  less  simple  than  satisfactory.  At  the 
same  time  the  determinations  which  produce  the  transition  from 
the  non-Euclidian  to  the  Euclidian  j^lanimetry,  are  shown  to  be 
identical  with  those  which  specify  the  surfaces  of  zero  curvature  in 
the  sanies  of  surfaces  of  constant  negative  curvature," 

Lobatchevski  and  Bolyai,  who,  together  with  Legendre,  were 
aware  of  the  defects  of  the  Euclidian  geometry  in  respect  to 
all  the  postulates  regarding  the  straight  line  and  the  plane,  have 


28 

both  tried  to  construct  them  and  to  deduce  their  properties 
from  the  construction.*  But,  as  it  seems  to  me,  they  did  not 
succeed  in  doiTig  it  with  sufficierd  rigor ;  and,  besides,  neither  of 
them  freed  himself  from  the  idea  that  a  plane  is  given  in  concep- 
tion previous  to  a  straight  line,  and,  therefore,  they  constructed 
first  the  plane  and  then  the  straight  line  in  it.  The  straight  line, 
therefore,  again  has  the  same  properties  as  a  geodesic  upon  a 
surface,  which  will  coincide  with  another  geodesic  in  a  similar 
surface,  of  same  constant  curvature,  as  soon  as  the  two  surfaces 
are  superposed  so  that  two  congruent  pairs  of  points  in  the  two 
geodesies  are  made  to  coincide.  At  least,  neither  of  the  two 
mathematicians  separated  the  straight  line  from  the  plane  suffi- 
ciently, to  come  to  the  clear  idea,  that  figures  of  straight  lines  in 
space  can  he  considered  without  considering  the  planes  in  which 
they  lie.  This  is  one  of  the  reasons  why  they  could  not  prove  the 
postulate  of  parallels,  which,  in  fact,  distinguishes  the  plane  from 
all  other  surfaces  of  constant  curvature.  For,  it  is  evident,  that 
since  the  curvature  of  a  surface  is  an  extrinsic  property  of  the 
surface,  i.  e.,  it  is  a  parameter  by  varying  which  we  can  obtain 
all  surfaces  of  constant  positive  and  negative  curvature,  the 
limit  between  the  two  being  the  plane  (of  zero  curvature)t,  — 
those  properties  of  the  geodesies  which  depend  upon  any  parti<M- 
lar  value  of  the  curvature,  could  not  he  found,  except  hy  leaving 
the  surface  and  going  out  into  space.  Of  course,  the  expression 
of  the  linear  element  can,  certainly,  give  the  true  metrical  prop- 
erties of  the  corresponding  surface,  and  hence,  also  of  the  plane. 
J3ut  this  very  expression,  as  Riemann  has  shoicn,  depends  upon 
the  curvature.  (In  fact,  for  a  surface  of  Gaussian  curvature, 
Ijh^  =  a,  the  linear  element  is  reducible  to  the  form 


i  +  |2- 


V^dc\ 


so  that  in  case  of  positive  curvature,  a  is  -f  ,  of  negative  curva- 
ture, a  is  —  ,  and  in  case  of  the  plane,  if  the  ^* parallel  postulate  " 

*See  "Urkunden  znr  Gesohichte  der  Nioht-Euklidischen  Geometric, 
Nikolaj  Iwanowitsch  Lobatschefskij, "  by  F.  Engel,  1898  (pp.  93-109),  and 
Frisohanf,  "  Elemente  der  absoluten  Geometric,"  1876,  pp.  8-18. 

t  dt  Ijk^  gives  curvature  of  spheres  and  pseudospheres,  with  all  their, 
varieties  resulting  from  bending,  and  k=  co  gives  the  plane.  See  Biemann, 
Ueber  die  Hypothesen,  etc.,"  II,  5. 


29 

holds,  a=  Oj  and,  conversely,  if  a  =  0,  then  the  postulate  of  par ' 
allels  holds.) 

Hence,  as  long  a^  we  remain  in  the  surface  itself,  and  suppos- 
ing  we  do  not  know  the  form  of  the  linear  element,  we  could  in  no 
way  prove  or  disprove  the  ^^  parallel  postulated'  And  this  is 
exactly  the  reasoning  of  those  who  think  that  the  "  parallel  postu- 
late "  cannot  be  proved,  I  could  not  do  better  than  refer 
again  to  the  last  quotations  from  Beltrami,  who,  although 
he,  as  far  as  seen  from  his  "  Saggio "  and  "  Teoria  fonda- 
mentale,''  may  never  have  expressed  himself  directly  on  the 
possibility  or  impossibility  of  proving  the  postulate  of  paral- 
lels, showed,  however,  the  reason  why  Lobatchevski  and 
Bolyai  had  arrived  at  a  geometry  for  the  plane  which  is  actu- 
ally true  only  for  pseudospheres.  In  another  place  in  his 
"  Saggio"  he  says  :  "We  see,  then,  that  two  points  of  the  sur- 
face (pseudospherical  surface),  chosen  in  any  manner  whatever, 
always  determine  uniquely  a  geodesic  line.  .  .  .  Thus,  sur- 
faces of  constant  negative  curvature  are  not  subject  to  excep- 
tions which,  in  this  respect,  happen  in  the  case  of  surfaces  of 
positive  curvature,  and,  therefore,  we  can  apply  to  them  the 
theorems  of  the  non-Euclidian  planimetry.  Moreover,  these 
theorems,  in  their  greatest  part,  are  not  susceptible  of  a  conc7^ete 
interpretation,  unless  they  are  referred  precisely  to  these  surfaces, 
instead  of  the  plane,  as  we  are  going  to  prove  presently  in  detaiV 

It  is  interesting  to  compare  this  sober  view  upon  the  non- 
Euclidian  geometry  of  the  great  Italian  mathematician,  whose 
works  more  than  those  of  any  other  succeeded  in  putting  this 
geometry  upon  a  respectable  footing,  with  a  few  quotations 
from  Bianchi,  "  Lezioni  di  geometria  differenziale,"  German 
translation,  1898,  t.  II,  p.  434.  The  quotations  referred 
to  are  headed  by  the  superscription,  "  A  Bev^iew  of  the  JVon- 
Euclidian  Geometry, '^  and  follow  an  excellent  account  of  pseu- 
dospherical geometry  treated  by  the  method  of  conform  repre- 
sentation, and  they  read  as  follows  :  — 

"  In  the  principal  theorems  of  the  pseudospherical  geometry, 
which  we  have  deduced  in  the  preceding  paragraphs,  a  close 
analogy  is  observable  to  the  propositions  of  the  plane  and  the 
spherical  geometry.  The  basis  for  these  analogies,  as  well  as 
for  the  differences,  we  can  foresee  a  priori.  If,  in  fact,  we 
examine  the  axioms  and  the  fundamental  postulates  of  the  plane 
geometry,  as  laid  down  by  Euclid  in  his  first  book,  and  if,  in  case 


30 

of  pseudospherical  surfaces^  we  replace  the  straight  line  by  the 
geodesic  line,  v:e  see  thai  when  we  leave  out  of  consideration  the 
XII  postulatej  concerning  parallels^  all  others  will  hold  without 
change  in  the  pseudospherical  geometry.  This  is,  in  particular, 
the  case  with  the  principle  of  congruence  and  also  ivith  tlie  prin- 
ciple that  a  geodesic  line  is  uniquely  determined  by  two  of  its 
points.  Those  propositions  of  plane  geometry  which  do  not 
depend  upon  the  parallel  postulate  hold,  therefore,  for  the  pseudo- 
spherical geometry ;  the  others  undergo  such  changes  as  to  become 
identical  with  the  old  ones,  as  soon  as  the  radius  It  of  the  pseudo- 
sphetical  surface  is  made  infinite, 

"  The  above  considerations  show  readily  the  uselessness  of 

ALL  ATTEMPTS    MADE   TO  PROVE  THE    PARALLEL    POSTULATE. 

If  this  proof  could  be  logically  deduced  from  the  other  principles, 

IT  WOULD  HAVE  TO  HOLD  EQUALLY  FOR  THE  PSEUDOSPHERI- 
CAL SURFACES  IN  EuCLIDIAN  SPACE. 

"  And,  in  fact,  if  in  the  plane  geometry  we  drop  the  Euclid- 
ian postulate,  we  are  led  to  the  so-called  abstract  or  non- 
Euclidian  geometry,  whose  foundations  were  laid  down  by 
Bolyai  and  Lobatchevski,  and  which  (the  straight  line  being 
taken  as  unlimited)  perfectly  coincides  with  the  pseudospheri- 
cal geometry." 

Now,  this  objection  is  certainly  valid,  if  we  were  bound  to  con- 
sider plane  figures  only.  Fortunately,  we  can  have  constructions 
involving  distances,  or  straight  lines,  and  angles,  nx)t  bound  to  lie 
in  any  particular  surface  at  all.  Tlie  conception  of  a  quadri- 
lateral without  fixed  area,  consisting  of  four  fixed  distances,  ivhich 
are  equal  by  pairs  (  opposite  sides ),  and  whose  angles  are  vari- 
able, and  equal  by  pairs,  is  an  important  conception  in  elementary 
geometry.  A  triangle  is  fixed  as  soon  as  its  sides  are  fixed : 
it  cannot  "  rack "  ;  its  area  and  its  angles  are  fixed  with  its 
sides  ;  it  is  necessarily  a  plane  figure.  And  if  the  proposition 
is  true  that  the  area  of  a  geodesic  triangle  =  a  const,  times  the 
excess  or  deficiency  of  the  sum  of  the  angles  of  the  triangle  over 
TT,  i.  e.,  A  =  db  K\A  -\-  B  -\-  C—ir),  where  A  is  always  posi- 
tive and  K  is  the  radius  of  curvature,  we  have  no  means  of 
varying  this  area,  when  the  geodesic  triangle  is  conceived  as  bound 
up  with  the  surface,  But,^  the  immaterial  quadrilateral,^  which 
consists  always  of  two  triangles,  is  not  bound  up  with  any  fixed 

*See  Definition  10  of  my  Dissertation  (p.  44). 


31 

area  and  is  not  bound  to  move  in  any  particular  surface^  and  its 
angles,  under  certain  imposed  conditions,  being  in  certain  deter- 
minate relations  among  themselves,  are  still  variable,  —  and^  by  conn 
sidering  the  continuous  series  of  deformations  given  in  Theorem 
18,"^  we  arrive  at  the  conxilusion  that  there  must  exist  a  quadrilat- 
eral with  equal  opposite  sides,  whose  two  angles  adjacent  to  the 
same  side  are  equal  to  two  right  angles,  not  knowing,  however, 
whether  the  quadrilateral  is  a  plane  quadrilateral  or  not.  And 
then  only,  with  the  aid  of  the  lemma.  Theorem  19,  which  proves 
that  the  sum  of  any  two  face-angles  of  a  triedral  angle  is  greater 
than  the  third,  we  find  either  that  such  a  quadHlateral  must  be  a 
plane  quadrilateral,  or  that  the  sum  of  the  three  angles  is  grecder 
than  two  right  angles.  But  the  second  alternative,  which  holds 
for  surfaces  of  positive  curvature,  was  proved  to  be  impossible  in 
the  case  of  a  plane,  by  Theorems  Hr-17;  hence,  only  the  first  altet^- 
native  remains,  from  which  it  immediately  follows  that 

THE  SUM  OF  THE  THREE  ANGLES  OF  A  RECTILINEAR  TRI- 
ANGLE IS  EQUAL  TO  TWO  RIGHT  ANGLES,  OR,  in  other  words, 

THE  GEO]METRY  OF  THE  PLANE  IS  PARABOLIC. 

Now,  it  so  happens  that  this  proof  has  been  formulated  in  such 
a  manner  as  to  take  in  consideration  the  existence  of  both  the 
spherical  and  the  pseudospherical  geometries.  It  actually  leaves 
room  for  the  existence  of  surfaces  of  constant  curvature — such  where 
the  sum  of  the  three  angles  of  a  geodesic  triangle  is  greater  than 
two  right  angles,  and  such  where  this  sum  is  less  than  two  right 
angles.  The  first  prove  to  be  finite  in  extent,  all  their  normals 
meeting  in  a  point  at  a  finite  distance,  —  as  follows  immedi- 
ately from  Theorems  14  and  17,  since  these  theorems  assume 
the  impossibility  for  two  geodesies  in  a  plane  to  intersect 
in  two  points,  and  the  infinite  extent  of  the  plane,  (both  of 
which  assumptions  have  been  proved  in  Theorems  2-5  and  8) ; 
while  the  second  class  of  surfaces  have  their  normals  non-co- 
planar,  or  meeting  at  an  imaginary  point.  Hence,  the  plane 
appears  to  be  the  limiting  case  between  the  two,  being  infinite 
in  extent  and  having  its  normals  meeting  in  a  point  at  oo.  The 
considerations  of  Bianchi  and  others,  which  tJiey  put  forward 
against  the  possibility  of  proving  the  Euclidian  postulate,  on  the 
ground  that  if  it  could  be  proved  for  the  plane,  it  would  have  to 
hold  for  the  pseudosphere  (or  for  the  sphere),  evidently  have  no 
value  against  such  a  proof  where  both  of  these  cases  have  been  con- 

*For  the  pages  corresponding  to  the  references  given  on  this  page  see 
Table  of  Contents,  above. 


32 

sidered,  mid  where  the  proof  first  establishes  the  existence  of  a 
kind  of  metrics  somewhere  in  space,  that  afterwards  is  proved  to 
he  possible  only  for  the  surface  of  zero  curvature  which  is  the 
plane.  Then  only  follows  the  proof  of  JEuclid^s  postulate  proper , 
for  the  plane,  which  by  its  metrics  is  now  singled  out  among  all 
surfaces  of  constant  curvature,  as  the  only  one  for  which  the  sum 
of  the  angles  of  a  triangle  equals  two  right  angles. 

If  we  examine  in  detail  the  axioms  which  have  been  made 
use  of  in  my  Dissertation,  we  find  that  they  are  actually  those 
which  are  necessary  for  defining  the  group  of  continuous  motions 
in  general,  without,  however,  postulating  that  a  point  in  space  is 
definable  by  three  coordinates.  Of  the  postulates  that  Helmholtz 
enumerates  in  his  work  "Ueber  die  Thatsachen,  die  der  Geo- 
metrie  zum  Grunde  liegen,"  Kon.  Ges.  der  Wis.  zu  Gottingen, 
1868,  I  assume  only  the  first  and  the  third,  stated  as  follows  : 

1)  Continuity  in  motion  —  defined  exactly  in  Axiom  1,  on 
page  63  of  my  memoir. 

2)  Free  mobility  of  rigid  bodies  and  consequent  independ- 
ence from  position  of  spacial  figures  of  all  kinds. 

To  these,  whose  order,  of  course,  is  reversed,  I  add  impene- 
trability and  infinite  divisibility.  The  number  of  dimensions  I 
do  not  assume,  but  deduce  from  the  postulates. 

A  word  still  with  regard  to  the  non-Euclidian  geometry. 
What  is  its  significance  and  the  place  it  has  to  occupy  in  the 
general  science  of  geometry  ? 

I  shall  here,  even  at  the  risk  of  repetition,  recapitulate  and 
supplement  in  a  concise  way  what  I  have  said  with  reference 
to  the  two  non-Euclidian  groups  of  motions  obtained  by  Lie 
from  the  Riemann-Helmholtz  axioms  as  modified  by  him. 
I  think  that  in  this  regard  the  position  of  Beltrami  in  his 
'^  Saggio  "  and  "  Teoria  fondamentale  "  is,  in  general,  the  only 
tenable  one.  In  our  point-space  of  three  dimensions  the  Lo- 
batchevskian  and  Riemannian  geometries  for  two  dimensions 
are  realized  respectively  on  the  pseudospherical  surfaces  and  on 
the  surfaces  of  constant  positive  curvature,  and  they  can  have  no 
other  concrete  interpretation.  With  special  conventions,  how- 
ever, as  to  the  meaning  of  ^'  distance  '^  and  '^  angle,''  we  get  the 
Poincar^  interpretation  by  means  of  what  he  calls  the  quad- 
ratic geometries,  i.  e.,  geometries  on  certain  quadric  surfaces. 
Professor  Klein's  interpretation  of  the  generalized  metrics  can 
in  no  way  make  the  plane  become  either  an  elliptic  or  a  hy- 
perbolic plane,  and  if  Cayley's  generalized  metrics  can  be  turned 


33 

into  account  for  obtaining  the  elliptic,  hyperbolic,  and  parabolic 
geometries,  it  is  only  for  different  surfaces  actually  existing  in 
the  Euclidian  space  that  this  interpretation  can  be  of  any 
value.  The  very  arbitrariness  of  these  different  kirids  of  metrics, 
which  depends  upon  the  arbitrary  value  of  the  constant  c  in  the 
formida  *  (x,  y)^  c  log  [.r,  y,  0,  0'~\y  where  0,  0'  are  the  points 
of  intersection  of  the  range  xy  with  the  fundamental  quadric, 
shows  tliat  the  different  metises  must  refer  to  a  whole  series  of  dif- 
ferent tivo-dimensional  manifolds,  differing  in  curvature  and  con- 
stituting the  elements  31^  of  some  R^  The  aggregate  of  all  these, 
however,  will,  in  our  case,  constitute  a  flat  manifold  of  three 
dimensions,  namely,  the  point-space  of  our  experience,  — just  as 
the  aggregate  of  all  possible  plane  curves  of  different  curva- 
ture passing  through  a  point  in  a  plane,  constitute  a  plane 
manifold  of  two  dimensions. 

As  to  the  non-Euclidian  metrics  in  three  dimensions,  I  can- 
not see  any  interpretation  for  this,  unless  the  space  to  which 
these  metrics  refer,  be  a  derivative  manifold  contained  in  a 
higher  manifold  of  four  dimensions,  since  again  the  very  para- 
meter of  the  curvature  suggests  only  a  particular  case  out  of 
an  infinity  of  possibilities,  arrived  at  by  giving  the  param- 
eter all  possible  values,  ranging  from  —  oo  to  -f  oo,  and  the 
aggregate  of  all  these  would  then  plainly  make  a  manifold  of  four 
dimensions.  Here,  again,  Beltrami  seems  to  have  hit  the  truth 
with  regard  to  this  interpretation  of  the  geometries  of  Riemann 
and  Lobatchevski  for  space  of  three  dimensions.  Thus,  at  the 
beginning  of  his  '^  Saggio,"  Beltrami  says  :  "  We  have  sought  to 
render  account  to  ourselves  of  the  results  to  which  this  new 
doctrine  (the  geometry  of  Lobatchevski)  leads ;  and  following 
a  process  which  seemed  to  us  actually  to  conform  to  the  good  tradi- 
tions of  scientific  investigation,  we  have  tmed  to  find  a  real  basis 
to  these  results.  We  think  we  have  found  one  for  the  planimetric 
par-t,  but  it  seems  to  us  impossible  to  find  one  for  the  case  of  three 
dimensions.''  And  then,  at  the  end  of  the  same  work,  after 
having  proved  the  interpretability  of  Lobatchevski's  plani- 
metry, in  every  particular,  by  the  geometry  upon  pseudospheri- 
cal  surfaces,  Beltrami  goes  on  to  say :  "  From  the  very  nature 
of  the  interpretation,  we  can  easily  foresee  that  there  can  exist  no 
analogous  interpretation,  equally  real,  for  the  non-Euclidian 
stereometry.     In   fact,  in  order   to  obtain    the   interpretation 

*Math.  Ann.,  Bd.  6. 


34 

which  we  have  just  given,  it  was  necessary  to  substitute  for  the 
plane  a  surface  which  cannot  be  reduced  to  a  plane,  that  is  to 
say,  whose  linear  element  can  in  no  way  be  reduced  to  the 
form  \/dx^  +  dy^y  which  essentially  characterizes  the  plane  itself. 
Consequently,  if  we  were  lacking  the  notion  of  surfaces  non-appli- 
cable  to  a  plane,  it  would  be  impossible  for  us  to  attribute  a  veri- 
table geometric  meaning  to  the  constructions  which  we  have  developed 
up  to  this  point.  Now,  the  analogy  leads  one  naturally  to  think 
that  if  there  can  exist  a  similar  interpretation  for  the  non-Euclidian 
stereometry,  this  interpretation  must  be  deducible  from  the  con- 
sideration of  a  space  whose  linear  element  is  not  reducible  to  the 
form  Vdx^  +  dy'^  +  dz^, — a  form  which  essentially  characterizes 
the  Euclidian  space.  And  since  up  till  now,  as  it  seems  to  us, 
we  have  been  wanting  the  notion  of  a  space  different  from  the 
Euclidian,  or,  at  least,  such  a  space  is  beyond  the  domain  of 
ordinary  geometry,  it  is  reasonable  to  suppose  that,  even  if  the 
analytical  considerations  upon  which  the  preceding  constructions 
are  based,  were  susceptible  of  being  extended  and  carried  over  from 
the  region  of  two  variables  into  that  of  three  variables,  nevertheless 
the  results  obtained  in  this  last  ca^e  could  not  be  interpreted  by  the 
ordinary  geometry.  This  conjecture  acquires  a  degree  of  prob- 
ability bounding  very  closely  on  certainty,  when  one  under- 
takes to  extend  the  preceding  analysis  to  the  case  of  three 
variables. 


"  Putting 


18 


E'  -"       -       -"      -"       ^       "V 


+  (a^  —  t^  —  u^)dv^  +  2uvdudv  +  2vtdvdt  +  2tudtdu'], 


which  takes  the  place  of  (1)  in  two  dimensions,*  — it  is  easy  to 
assure  oneself  that  the  analytic  deductions  obtained  from 
formula  (1)  subsist  integrally  for  the  new  expression,  and  that 
the  value  of  ds  given  by  this  last  is  effectually  the  value  of  the 
linear  element  of  a  space  in  which  the  non-Euclidian  (Loba- 

*  The  formula  (1 ),  here  referred  to,  is  the  one  which  Beltrami  gives  at  the 
beginning  of  his  work,  for  the  linear  element  of  a  surface  of  constant  negative 

curvature, ds^  ■=  W j-z — ^^ '- — ;  this  serves  as 

ya''  —  w-  —  v^Y 
the  basis  for  his  interpretation. 


35 

tchevsJdan)  stereometry  finds  an  interpretation,  as  completey  speak- 
ing ANALYTICALLY,  OS  that  given  for  the  planimetry, 
"  But  putting 

t=:  r  cos  p^j  u  =  r  sin  p^  cos  p^j  v  =  r  sin  p^  sin  p^, 
and 

Radr 

we  get 

ds'  =dp'-{-(R  sinh  ^  \\dpl  +  sin^  p^dp^, 

a  formula  which  shows  that  /),  p^,  p^  are  the  orthogonal  curvi- 
linear coordinates  of  the  space  considered. 

"  Now,  M.  Lam6  has  proved  that,  taking  as  curvilinear  co- 
ordinates of  points  in  space  the  parameters  p,  p^y  p^,  of  three 
families  of  orthogonal  surfaces  —  in  which  case  the  distance  be- 
tween two  infinitely  near  points  is  represented  by  an  expression 
of  the  form 

ds"  =  H^p"  +  H\dp\  +  Hldpl— 

the  three  iJ^s,  as  functions  of  the  /j's,  must  satisfy  two  distinct 
systems,  each  consisting  of  three  partial  differential  equations 
of  the  types, 

^    _  _L      ^       ^         J_      ^      ^2 

and 

dp,  Ui  ^j  +  ^p,U2  ^P2r H^  ^p  >  ^^ 

(Le9ons  sur  les  coordin^es  curvilignes,  pp.  76  and  78). 
"  In  our  case 

H—  ly  H,  —  R  sinh  -^,  H^  —  R  sinh  -^  sin  p, , 

and  for  these  values,  the  first  system  is  evidently  satisfied  ;  but 
the  second  system  is  satisfied  only  for  R=  oo.  Hence  the 
expression  (18)  cannot  belong  to  the  linear  element  of  the  ordi- 
nary Euclidian  space,  and  the  formulce  founded  upon  this  exp^-es- 


36 

slon  cannot  be  constructed  by  means  of  figures  given  us  by  the  or- 
dinary geometry.'^ 

And,  again,  in  his  "  Teoria  fondamentale  degli  spazii  di  cur- 
vatura  costante/'  Beltrami  says  : 

"  Thus  all  conceptions  of  the  non-Euclidian  geometry  find 
a  perfect  correspondence  in  the  geometry  of  a  space  of  constant 
negative  curvature.  It  is  only  necessary  to  observe  that  while 
the  conceptions  of  the  planimetry  obtain  a  true  and  proper  inter- 
pretatioUy  since  they  can  be  constructed  upon  a  real  surface,  those, 
on  the  contrary,  which  refer  to  three  dimensions,  are  susceptible 
only  of  an  analytic  representation,  since  the  space  in  which  such 
a  representation  could  be  realized  is  different  from  that  to  which 
we  ordinarily  give  the  name  of  space.  At  least,  it  does  not  seem 
that  experience  could  be  brought  into  agreement  with  the  results 
of  this  more  general  geometry,  unless  we  suppose  R  to  be  infi- 
nitely great,  that  is,  the  curvature  of  the  space  to  be  zero. 
This  circumstance  might,  of  course,  also  be  due  only  to  the 
smallness  of  the  triangles  which  we  can  measure,  or  to  the 
smallness  of  the  region  to  which  our  observations  extend  them- 
selves.'^ 

In  his  "Teoria  fondamentale,^'  Beltrami  shows,  from  a  general 
discussion  of  n-dimensional  manifolds,  that  the  linear  element 
in  the  Riemannian  geometry  of  three  dimensions  may  be  taken 
to  be  the  same  as  the  linear  element  upon  a  hypersphere  in  a 
space  of  four  dimensions.  The  equation  of  the  hypersphere, 
with  center  at  origin,  will  be 

fj^u'  +  v^JrW^^  a\ 
and  hence, 

da^  =  de  +  du^  -f  dv^  +  dw" 

is  at  once  the  representation  of  the  linear  element  upon  the  hyper- 
sphere of  radius  a,  and  in  a  Riemannian  space  of  curvature  1  ja^. 
To  obtain  the  linear  element  of  the  three-dimensional  space 
of  Lobatchevski,  he  substitutes  ds—  —  Eda/w,  and  by  elimi- 
nating 10  he  gets  (18).  The  curved  Lobatchevskian  space,  of 
infinite  extent,  is  then  imaged  upon  the  interior  of  the  sphere 
of  Euclidian  space, 

e-^u'  +  v^^  a\  — 

the  geodesies  of  that  space  being  represented  by  chords  of  the 


37 

sphere.  Every  geodesic  has  two  distinct  real  points  at  oo,  which 
are  imaged  upon  the  representative  sphere  by  the  two  ends  of 
the  corresponding  chord,  so  that  the  spherical  surface  itself  cor- 
responds to  the  locus  of  all  points  at  oo  in  the  Lobatchevskian 
space. 

But  since  we  proved  *  that  to  assume  a  four-dimensional 
point-space  is  to  commit  a  logical  error,  and  since  Beltrami's 
results  have  certainly  given  a  conclusive  analytical  proof  that  we 
could  obtain  Lobatchevski's  geometry  for  three  dimensions,  if 
we  could  actually  construct  a  curved  three-dimensional  manifold, 
contained  in  a  four-dimensional  plane  manifold,  —  we  may  sur- 
mise that  the  only  way  to  obtain  a  concrete  and  true  interpreta- 
tion of  the  Lobatchevskian  (as  well  as  the  Riemannian)  stere- 
ometry is  to  be  found  in  Pliicker's  idea  that  our  space  becomes  a 
manifold  of  a  higher  number  of  dimensions,  when,  instead  of  the 
point,  we  take  as  its  element  a  figure  depending  upon  n  para- 
meters, making  space  a  manifold  of  n  dimensions.  Therefore, 
it  would  seem  that  one  of  the  simplest  ways  to  look  for  such  a 
concrete  interpretation  would  be  to  start  with  line  geometry, 
which  makes  space  an  H^  —  the  next  simplest  after  the  point- 
space,  which  is  an  R^  —  and  seek  what  in  this  geometry  would 
be  meant  by  the  terms  :  "  distance,''  "  angle,"  ^'  linear  element," 
'^curvature,"  "parallel,"  "perpendicular,"  and  other  metrical 
terms  ;  and  see  whether  the  results  thus  arrived  at,  —  by  con- 
sidering in  it  the  special  three-dimensional  manifolds  (com- 
plexes) possessed  of  "  curvature,"  (since  by  excluding  the  postu- 
late of  parallels,  which  is  now  proved  for  a  flat  manifold, 
such  as  our  point- space  must  undoubtedly  be,  we  actually  ob- 
tain a  manifold  of  constant  curvature),  —  whether  these  results 
do  agree  with  those  obtained  in  the  Lobatchevskian  and 
Riemannian  geometries,  respectively.  But  such  an  investiga- 
tion would  go  far  beyond  the  limits  of  the  present  Dissertation. 

*  In  the  introductory  chapter  of  the  Dissertation. 


ON    THE   FOUNDATIONS  OF   THE   EUCLIDIAN 
GEOMETEY. 


Chapter  I. 

SPACE  AND  ITS  DIMENSIONS. 


Definition  1.  —  Geometry  is  the  science  which  treats  of  spacdal 
forms  and  magnitudes  and  their  mutual  relations.  Dealing 
with  magnitudes,  and  with  spacial  forms  only  in  so  far  as  these 
are  determined  by  their  magnitudinal  relations,  Geometry  is  a 
branch  of  the  general  science  of  quantity  —  Mathematics, 

A  few  introductory  remarks  are  necessary  in  the  way  of  more 
accurately  specifying  the  subject  of  geometry,  which  I  prefer 
to  put  in  the  form  of  definitions.  These  definitions,  however, 
will  not  be  only  nominal ;  most  of  them  prove  the  actual  ex- 
istence of  the  objects  they  define. 

Definition  2.  —  Space  is  that  in  which  all  bodies  exist.  It  is 
the  condition  sine  qua  non  of  material  objects.  This  truth  is 
expressed  in  physics  by  the  assertion  that  matter ,  or  the  sub- 
stance of  which  all  bodies  consist,  has  extension,  or,  in  other 
words,  material  objects  occupy  space. 

Definition  3.  —  Experience  teaches  us  that  matter  is  also  im- 
penetrable, i.  e.,  that  every  material  object  occupies  a  definite  por- 
tion of  space,  which  is  fixed  by  certain  limits  or  boundaries  and 
tohich  cannot  at  the  same  time  be  occupied  by  any  other  material 
object.  The  portion  of  space  that  is  for  a  time  exclusively  occupied 
by  a  certain  material  object  is  called  the  place  of  that  object. 
For  the  sake  of  accurate  terminology  I  propose  to  call  it  the 
geometrical  place. 

Definition  4.  —  Experience  further  teaches  us  that  the  re- 
sources of  space  with  regard  to  its  capacity  of  containing  ma- 
terial objects,  or  of  affording  place  to  the  material  substance, 
are  absolutely  limitless.  Thus,  notwithstanding  the  impenetra- 
bility of  material  substance,  explained  above,  beside  any  occu- 
pied space  there  is  always  room  enough  for  the  existence  of  other 
material  objects,  and  any  vacant  space  is  always  conceived  of  only 
as  susceptible  of  being  filled  up  with  matter.  Moreover,  any 
material  object  is  conceived  of  as  capable  of  being  divided  into 

38 


39 

any  number  of  portions,  and  these  again  subdivided  into  lesser 
ones,  and  so  on,  ad  infinitum.  In  this  respect,  extended  sub- 
stance  and,  hence,  also  space,  follow  perfectly  the  nature  of  ah~ 
stract  quantity,  ranging  both  ways  —  from  the  finite  to  the  in- 
definitely small,  on  the  one  hand,  and  to  the  indefinitely  large, 
on  the  other. 

Definition  5.  —  The  geometrical  place  of  a  body,  being  that 
portion  of  space  which  is  occupied  by  that  body  to  the  exclusion  oj 
any  other  body,  lias  the  same  spacialform  and  dimensions  as  the 
body  which  fills  it  up.  We  mean  by  this,  that  whatever  meas- 
urements in  regard  to  extension  the  whole  body,  or  its  several 
parts,  may  have,  the  same  are  attributed  to  its  geometrical 
place,  and  whatever  arrangement  of  extended  parts  makes  up 
the  form  of  the  body,  the  same  belongs  likewise  to  the  geomet- 
rical place,  and  vice  versa,  —  so  that  in  these  regards  the  geomet- 
rical place  may  be  substituted  for  the  body,  and  conversely. 
The  geometrical  place  alone,  apart  from  all  other  physical  prop- 
erties, or,  in  fact,  apart  from  the  matter  filling  it  up,  is  dealt 
with  in  geometry,  —  and  is  regarded  by  this  science  : 

1°.  As  a  magnitude — that  is,  not  only  as  something  that 
can  be  greater  or  less,  the  reason  for  which  is  given  in  Definition 
4,  but  as  something  that  can  be  measured,  that  is  accurately 
compared,  with  a  view  of  an  exact  quantitative  determination, 
with  a  standard  mxxgnitude  of  the  same  kind,  which  is  arbi- 
trarily taken  as  a  unit,  and  can  be  repeated  any  number  of 
times,  or  divided  into  a  certain  number  of  equal  parts,  thereby 
becoming  either  equal  to,  or  greater,  or  smaller  than  the  mag- 
nitude in  question ;  and 

2°.  As  a  form,  consisting  of  a  definite  arrangement  of  parts 
according  to  some  law,  which  can  also  be  expressed  by  numbers. 

Definition  6. — The  geometrical  place  of  a  body  is  called  a 
solid  in  geometry,  meaning  by  it,  that  it  is  mentally  represented 
as  preserving  a  fixed  form  and  dimensions.  The  geometrical 
solid  is  a  mere  ideal  abstraction  and  has  nothing  to  do  with  phys- 
ical solidity,  from  which,  however,  it  is  originally  derived. 
Geometry  does  not,  therefore,  treat  it  as  impenetrable.  It  is, 
indeed,  only  the  impression  left  by  a  body  in  surrounding  space 
conceived  of  as  capable  of  preserving  the  impression  after  the 
body  itself  has  been  removed.  As  a  magnitude  or  thing  to  be 
measured  and  expressed  in  numbers,  mthout  regard  to  its 
form  or  outer  appearance,  it  is  called  volume. 


40 

Postulate  1. —  The  geometrical  solid  or  body  may  be  mentally 
imagined  as  moving  about  in  fixed  space,  or  changing  its  position 
with  respect  to  other  bodies,  whether  physical  or  geometricaly 
without  distortion  or  change  of  form.  The  solid  is  said  to  pos- 
sess geometrical  rigidity,  meaning  by  it  that  the  disposition  of  the 
parts  with  reject  to  each  other  is  fixed  and  unchangeable,  or,  that 
there  is  no  internal  motion.  This  idea  of  geometry  is  derived 
from  the  fact  that  space  is  conceived  of  as  affording  a  mere  passive 
capacity  of  being  filled  up  with  matter,  all  changes  of  form  being 
r^erred  to  the  active  principle  of  the  material  substance  py^oper, 
i.  e.,  to  physical  causes  alone  (of.  Definition  4).  It  is  also  based 
upon  the  undoubted  fact  of  universal  experience  that,  in  so 
far  as  can  be  ascertained  by  observation  and  experiment 
(measurements  —  astronomical,  physical,  and  geodesic),  no  real, 
or  physical,  ligid  body,  moving  about  in  space,  has  ever  been 
known  to  undergo  any  alteration  in  form  or  dimensions  on  account 
only  of  change  of  position  in  space,  without  regard  to  physical 
causes  which,  in  most  cases,  have  been  found  quite  adequate  to 
account  for  such  alterations.  And  even  if  the  contrary  were 
the  truth  in  the  case  of  real  bodies,  the  Euclidian  geometry 
would  still  have  nothing  to  do  with  such  alterations,  as  it  con- 
siders only  ideal  rigidity,  where  change  of  form  or  dimensions 
as  depending  upon  position,  is  purposely  eliminated  for  the 
sake  of  simplicity,  and  may  be  left  to  other  branches  of  the 
mathematical  sciences  to  consider  (Kinematics,  for  instance, 
may  very  properly  consider  such  questions  as  a  special  kind  of 
liaisons  —  constraint  —  depending  upon  any  number  of  para- 
meters, those  of  position  included).  But  the  Euclidian  geom- 
etry considers  only  the  simplest  case,  even  if  it  were  only  an 
idealized  abstraction. 

Definition  7.  —  A  body  is  said  to  be  equal  to  another  geomet- 
rically, when  their  geometrical  places  can  be  made  to  fill  each 
other  without  remainder  of  any  parts  of  the  one,  not  filled  by 
corresponding  parts  of  the  other.  When  the  geometrical  places 
thus  fill  each  other,  we  say  that  the  geometrical  bodies  coincide, 
—  coincidence,  as  thus  defined,  being  a  proof  of  equality,  in- 
variably resorted  to  in  geometry. 

When  the  coincidence  can  take  place  only  with  the  change 
of  form  resulting  from  a  mere  rearrangement  of  parts,  these 
last  preserving  separately  their  respective  magnitudes  and  forms, 
the  bodies  are  said  to  be  of  equal  volume,  though  not  equal  in 
form. 


41 

Definition  8.  —  One  body  is  said  to  be  r/reater  than  another, 
when  some  of  the  parts  of  the  one  can  be  made  to  coincide 
with  all  the  parts  of  the  other,  while  there  still  remain  some 
parts  of  the  first,  having  no  corresponding  parts  of  the  second 
to  coincide  with.     The  other  body  is  then  called  the  less. 

Scholium.  —  Having  firmly  established  the  empirical  and 
rational  basis  of  the  notions  contained  in  the  previous  defini- 
tions, it  may  not  be  amiss  to  give  a  more  compact  and  abstract 
form  to  the  logical  process  by  which  they  are  obtained,  which  is 
free  from  all  cavils  on  the  part  of  those  who  think  that  spacial 
forms  and  magnitudes  may,  for  all  we  know,  be  certain  functions 
of  absolute  position,  which  we  shall  never  be  able  to  ascertain  or 
disprove.  Starting  with  the  notions  of  space,  matter  or  ex- 
tended substance  in  general,  position,  and  change  of  position  or 
motion,  and  with  the  abstract  notion  of  quantity,  we  may  as- 
sume, for  the  sake  of  abstraction,  the  existence  of  a  hypothet- 
ical impenetrable  material  substance,  infinitely  divisible,  —  i.  e., 
possessed  of  the  following  properties  :  — 

1 .  Impenetrability.  —  Every  determiiiate  portion  or  quantity  of 
this  substance  occupies  or  fills  up  a  corresponding  portion  of  space 
which  cannot  at  the  same  time  be  occupied  by  any  other  portion  of 
the  same  substance.  Any  tioo  portions  of  space  thus  filled  up  by 
the  same  quantity  of  the  substance  at  different  times ,  are  said  to  be 
equal  in  capacity y  and  any  two  portions  of  the  substance  tohich 
can  fill  up  the  same  portion  of  space  at  different  times,  or  dif- 
ferent portions  of  space  of  equal  capacity  at  the  same  time,  are  said 
to  be  of  the  same  bulk.  So  that  to  each  bulk,  which  measures 
the  quantity  of  the  hypothetical  substance,  there  is  a  correspond- 
ing capacity  of  the  space  which  is  occupied  by  it  at  any  moment, 
to  the  exclusion  of  any  other  portion  of  the  same  substance ; 
to  a  greater  portion  of  the  substance,  there  corresponds  a  greater 
capacity  of  the  space  occupied  by  it,  to  a  double  or  multiple 
bulk,  a  double  or  equimultiple  capacity  of  the  space,  and  to 
any  part  of  a  given  bulk,  a  corresponding  part  of  the  capacity 
of  the  space  occupied.  The  generic  term  for  bulk  or  capacity 
alike  is  volume,  so  that  the  quantity  of  the  substance  (bulk)  and 
the  spac£  filled  up  by  it,  to  the  exclusion  of  any  more  of  the 
same  substance  (capacity),  are  said  to  be  equal  in  volume. 

2.  Infinite  divisibility.  —  If  a  portion  of  the  substance  is  di- 
vided into  n  portions,  such  that  they  can  fill  up  spaces  of  equal 
capacity  each,  that  portion  is  said  to  be  divided  into  n  equal  parts. 


42 

The  property  of  infinite  divisibility  now  becomes  perfectly  com- 
prehensible, and  is  possessed,  according  to  hypothesis,  both  by 
the  hypothetical  substance,  and  by  the  spa^e  giving  position  to  this 
substance. 

3.  Form  ;  rigidity,  or  plasticity.  —  If  two  equal  poiiions  of 
the  substance  {of  equal  bulk  or  volume)  are  made  to  fill  up  succes- 
sively the  same  fixed  portion  of  space  (not  merely  spaces  of  equal 
capacity),  then  in  these  two  portions  of  the  substance  we  observe  not 
only  equality  in  bulk  of  the  whole,  but  also  of  corresponding  parts, 
filling  up  in  the  two  cases  the  same  corresponding  parts  of  space ; 
that  is,  the  two  portions  of  the  substance,  while  each  fills  up 
in  tmm  the  space  considered,  have  a  similar  arrangement  of  parts 
that  are  equal  in  bulk  in  the  two  cases,  no  mutter  in  what  man- 
ner the  division  is  made,  and  however  small  the  parts  considered. 
The  two  equal  portions  of  the  substance,  in  their  successive  positions, 
are,  therefore,  said  to  be  equal  not  only  in  volume,  but  also  in 
form  —  equal  form  thus  meaning  an  equal  arrangement  of  equal 
parts.  When  a  portion  of  the  substance  leaves  a  certain  posi- 
tion, passing  into  another  position,  it  may  change  its  form  (i.  e., 
the  arrangement  of  parts  may  change,  so  that  an  arrange- 
ment denoted  by  a,  b,  c,  •  -  -  k,  may  now  have  to  be  repre- 
sented by  •  •  •  e  "  •  k  .../...  g  ...  a  •  •  •  h  -  --,  where  a, 
^j  ^j  ^j  ff  9)  hf  h  ^tc.,  denote  unequal  portions).  Moreover, 
even  if  as  a  whole  it  does  not  change  its  position,  that  is, 
if  some  one  part  of  it,  at  least,  preserves  its  old  position,  the 
form  of  the  whole  may  still  change,  and  actually  does  change, 
whenever  the  remaining  parts  change  their  relative  positions  to  this 
fixed  part  and  to  each  other ;  and,  provided  the  space  filled  up 
by  the  whole  is  still  continuous,  that  is,  it  is  still  filled  up  com- 
pactly and  represents  one  concrete  whole,  without  interruptions 
of  vacant  or  unoccupied  portions  intervening,  we  say,  the  space 
occupied  by  the  whole  has  not  changed  its  volume,  but  has 
changed  its  form,  and  so  has  the  substance  filling  it  up.  The 
distinction  between  two  portions  of  space  of  equal  volume  and 
two  portions  of  both  equal  volume  and  equal  form  is  now  clear 
and  unambiguous.  In  fact,  we  have  seen  that,  when  the  sub- ' 
stance  as  a  whole  does  not  leave  its  original  position,  i.  e.,  when 
at  least  one  of  its  portions  preserves  its  position,  a  change  in  form 
is  possible  only  in  virtue  of  the  change  of  position  of  the  remain- 
ing parts  with  respect  to  the  stationary  part  and  with  respect  to 
one  another,  —  in  other  words,  change  of  form  is  caused  by  mo- 


43 

tion  of  parts  of  the  whole  with  respect  to  one  another,  that  is, 
by  internal  motion.  The  same  is,  therefore,  true  with  respect 
to  a  substance  which  has  left  its  original  position  entirely,  fill- 
ing now  up  a  portion  of  space  which  has  not  the  smallest  part 
in  common  with  the  original  position.  If  there  has  been  inter- 
nal motion  or  a  rearrangement  of  unequal  parts  besides,  then 
the  substance  has  also  changed  its  form  ;  if  there  has  been  no 
internal  motion,  the  original  arrangement  of  parts  having  been 
preserved,  the  substance  has  only  changed  its  position,  but  not 
its  form.  If  a  substance  resists  a  change  of  form,  as  just  de- 
fined, whether  the  whole  is  at  rest  or  in  motion,  we  may  say 
that  the  parts  are  held  fixed  to  one  another,  and  we  call 
this  state  of  the  substance  the  rigid  state ;  the  whole  substance 
is  then  said  to  form  a  solid  or  a  rigid  body.  The  portions  of 
space  which  represent  any  two  successive  positions  of  a  solid 
in  motion,  are  said  to  be  equal  to  each  other,  in  volume  and 
form,  just  as  two  solids  that  can  be  made  to  fill  up  successively 
the  same  space,  are  themselves  said  to  be  equal  in  volume  and 
form.  When  two  equal  solids  are  made  successively  to  fill  up 
the  same  space,  then,  by  abstracting  from  time,  that  is,  disre- 
garding the  fact  that  the  filling  up  can  take  place  only  at  dif- 
ferent moments,  we  simply  say  that  the  two  solids  are  made  to 
fill  up  the  same  space,  or,  they  are  made  to  coincide  with  each 
other  —  coincidence  being  a  test  of  equality  in  volume  and  form. 
If,  on  the  contrary,  the  substance  does  not  resist  a  rearrange- 
ment of  parts,  these  parts  are  not  held  rigidly  to  one  another, 
and  change  of  form  is  possible  without  change  of  volume. 
Such  a  portion  of  the  substance  is  said  to  possess  plasticity. 

We  see  now  that  these  notions,  though  having  a  firm  empir- 
ical basis,  are  not  absolutely  dependent  upon  the  actual  con- 
dition of  things.  The  hypothetical  substance,  of  absolute  impene- 
trability, need  not  actually  exist,  but  a^  an  abstraction,  agreeing, 
in  general,  with  our  expet^ence,  it  may  serve  as  a  starting  point 
for  the  only  possible  science  of  measurement  of  extension  ;  since  the 
notions  based  on  its  assumption  are  clear  and  unequivocal,  and 
absolutely  necessary  to  make  the  investigation  of  the  laws  of 
spacial  forms  and  magnitudes  possible.  Moreover,  we  must 
agree  to  class  all  actual  phenomena,  in  so  far  as  they  conform 
to  the  laws  deducible  from  these  notions  and  from  those  that 
follow  in  this  introduction,  as  geometrical  phenomena,  that  is, 
such  as  depend  upon  the  essence  of  extension  only ;  and,  in  so 


44 

far  as  they  deviate  from  these  laws,  they  must  be  explained  by 
physical  causes,  and  any  attempt  to  confuse  these  two  (as  very 
able  geometers,  like  Clifford  and  others,  have  done)  would 
only  tend  to  raise  a  dust  of  endless  discussion,  which  would 
never  permit  us  to  see  the  real  foundations  of  geometry. 

Definition  9.-^  A  rigid  physical  body  is  said  to  be  sur- 
rounded by  vacant  space  on  all  sides,  when  it  can  be  moved  in 
all  directions  :  forwards  and  backwards,  to  the  right  or  to  the 
left,  and  so  on,  in  all  intermediate  directions.*  JVhen  some 
other  body  is  posited  beyond  the  vacant  space,  in  any  part  of  it, 
the  two  bodies  are  said  to  be  at  a  distance  from  each  other,  that 
admits,  either  of  the  position  of  some  third  body  between  them, 
or  of  the  motion  of  one  towards  the  other.  In  the  latter  case, 
the  bodies  are  said  to  approach  each  other,  the  distance  between 
them  becoming  less  and  less,  until  it  vanishes  altogether,  ad- 
mitting of  no  further  approach  towards  each  other,  the  bodies 
then  being  in  contact.  These  ideas  of  distance  and  contact  are 
transferred  upon  geometrical  solids,  or  the  geometrical  places  of 
the  bodies  corresponding  to  the  positions  of  the  physical  bodies 
just  mentioned. 

Definition  10.  —  When  two  rigid  physical  bodies  are  brought 
into  close  contact  with  each  other,  so  that  no  further  motion  of 
one  toward  the  other  is  possible,  they  are  said  to  have  reached 
the  limits  or  boundaries  of  each  other,  and  if  these  limits  are  to 
some  extent  continuous,  —  i.  e.,  when  they  touch  each  other  in 
many  parts,  the  touchings  being  uninterrupted  by  intermediate 
vacant  space,  which  happens  when  the  bodies  fit  each  other, — the 
limits  are  then  called  surfaces.  A  physical  surface  is,  accord- 
ingly, the  continuous  boundary  where  the  rigidity  of  a  body 
just  begins,  or  where  the  physical  property  of  impenetrability 
just  begins  to  act.  If  the  body  is  surrounded  by  vacant  space, 
the  surface  of  the  body  is  the  boundary  separating  the  impene- 
trable matter  of  the  body  from  the  capacious  space.  But  it  is 
neither  the  one  nor  the  other,  since  the  smallest  part  of  the 
body  has  some  of  its  parts  removed  from  the  boundary  by  the 
interposed  rigidity  of  other  parts  of  itself.  No  other  rigid  body 
could  possibly  have  access  to  those  concealed  parts  without 
overcoming  rigidity,  and,  therefore,  no  part  of  the  body,  how- 

*  The  word  direction  is  used  here  in  the  common  acceptance  of  its  meaning, 
viz:  some  course,  but  it  is  really  vague.  The  scientific  meaning  of  the 
word  will  be  given  in  another  place  in  this  work. 


45 

ever  small,  can  belong  to  its  surface,  of  which  the  essential 
characteristic  is  its  being  in  contact  with  some  other  body,  w, 
ivith  vacant  space  and,  hence,  capable  of  being  brought  into 
contact  with  some  other  body.  Surface,  therefore,  has  no  mag- 
nitude of  the  same  kind  as  a  body  ;  in  other  words,  it  has  no 
bulk  or  volume,  and  it  can  never  amount  to  any  part  of  volume. 
But,  as  is  shown  in  the  following  definition,  it  nevertheless  has 
magnitude  and  form  of  its  own ;  in  other  words,  it  is  a  thing 
that  can  be  measured  and  expressed  in  numbers — these  numbers 
being  in  determinate  relations  to  those  expressing  the  volume 
of  the  body  bounded  by  the  surface.  One  of  the  tasks  of 
geometry  is,  in  fact,  the  discovery  and  determination  of  these 
relations.  The  idea  of  surface  is  also  transferred  from  physical 
bodies  to  geometrical  solids,  and  the  geometrical  surface  may  be 
said  to  represent  the  geometrical  place  ofaphydcal  surface.  Geom- 
etry regards  it  as  a  separate  entity,  capable  of  existing  by 
itself  and  moving  about  in  space,  or  changing  its  position  with 
regard  to  other  bodies  and  surfaces,  whether  physical  or  geo- 
metrical, without  distortion  or  change  of  form. 

Definition  11.  —  Since  a  rigid  body,  immersed  in  unoccupied 
space,  or  in  any  plastic  material  substance,  displaces  a  portion 
of  the  material,  or  occupies  a  portion  of  the  void,  to  the  exclu- 
sion of  other  matter,  equivalent  to  its  own  volume,  and  since 
this  rigid  body  exposes  only  its  surface — the  interior  parts  not 
coming  into  play  at  all  in  the  act  of  this  displacement  (the  in- 
terior might  as  well  be  imagined  hollow  or  devoid  of  matter  in 
this  connection), — it  is  evident  that,  in  general,  surface  ought  to 
be  a  function  of  volume,  increasing  with  the  increase  of  the 
last ;  that  is,  to  a  large  volume  there  must  in  general  correspond 
a  large  surface,  although  the  converse  is  not  a  necessary  conse- 
quence. At  any  rate,  it  is  quite  inconceivable  how  a  rigid  im- 
penetrable body  could  take  up  space  to  the  exclusion  of  other 
material  substance,  which,  on  account  of  its  capability  of  mo- 
tion or  change  of  position,  can  be  prevented  from  occupying  the 
same  space  as  the  body  consider ed,  only  by  the  boundaries  of  the 
last  J  — were  it  not  that  these  boundaries  are  in  themselves  an  extended 
magnitude,  standing  in  some  functional  relation  to  volume  and 
form.  Accordingly,  surface  must  have  portions,  all  of  which 
may  be  exposed  to  vacant  space,  or  in  contact  with  surfaces  of 
other  bodies,  or  some  exposed  and  the  others  covered  by  cor- 
responding surfaces  of  other  bodies.     In  this  last  case,  the  ex- 


46 

posed  surfaces  can  again  be  brought  into  contact  with  surfaces  of 
other  bodies.  Two  surfaces  will  then  be  equal,  if  they  can  lie 
upon  each  other  and  mutually  cover  all  their  parts ;  and  one  is 
greater  than  the  other,  when  a  part  of  the  first  can  cover  the 
whole  of  the  second,  while  another  part  of  the  first  will  remain 
exposed,  or  covered  by  a  third  surface. 

Scholium.  Any  part  of  a  body  may  be  regarded  as  a  sepa- 
rate body  (in  the  geometrical  sense  of  the  term)  from  the  re- 
maining part,  since  each  can  be  imagined  to  move  about  in 
space  independently  of  the  other,  and  without  distortion  or 
change  of  magnitude ;  and,  while  the  two  constitute  the  parts  of 
the  same  solid,  the  limit  common  to  both  is  a  surface  of  some 
definite  shape  and  magnitude. 

Corollary.  Surfaces  coincide  with  one  another  when  the 
bodies  limited  by  them  coincide  and,  conversely,  bodies  limited 
by  surfaces  that  can  be  made  to  coincide  with  one  another, 
must  themselves  be  capable  of  coincidence,  since  when  the  sur- 
faces are  brought  into  actual  coincidence,  none  of  the  bodies 
can  help  being  everywhere  within  and  nowhere  beyond  their 
coinciding  limits. 

Definition  12.  —  Both  experience  and  reasoning  lead  us  to 
the  conclusion  that,  while  volume,  or  unspecified  space — space  in 
all  possible  directions ,  wherever  motion  is  possible —  is  homogeneous, 
surface,  or  that  which  limits  a  body,  may  be  of  very  different 
kinds,  having  almost  nothing  in  common,  except  that  an  in- 
definitely small,  or  infinitesimal,  part  of  any  surface  may  be 
imagined  to  move  towards  another  infinitesimal  part  of  the 
same  surface  by  a  continuous  infinity  of  paths  in  the  surface 
itself,  as  will  be  shown  later.  But  this  common  property  is  not 
sufficient  to  make  surface  a  magnitude,  always  definable  with 
mathematical  precision,  and  capable  of  being  expressed  in  a 
voluntarily  chosen  unit.  The  indefinite  size  of  the  small  part 
that  is  capable  of  congruence  would  make  the  computation  of 
areas  with  mathematical  precision  impossible,  unless  there  be  ways 
of  reducing  these  to  surfaces  capable  of  coincidence  in  finite  por- 
tions. Volume,  as  a  magnitude,  which,  in  fact,  is  only  the 
capacity  of  space  to  contain  matter  of  a  constant  ideal  impene- 
trability, is  everywhere  the  same.  Any  part  of  volume  is 
capable  of  coincidence  with  any  other  corresponding  part  of 
volume ;  this  coincidence  is  given  directly  in  the  fundamental 
idea  of  motion  together  with  the  idea  of  rigidity  of  the  moving 


47 

bodies.  Volume,  therefore,  or  space  unspecified,  is  homo- 
geneous ;  whereas  surface,  as  having  an  infinite  variety  of  forms, 
is  not  so.  And  while,  for  instance,  a  smaller  body,  being  posited 
within,  or  surrounded  by,  a  larger  one,  invariably  occupies  a 
part  of  the  volume  of  the  larger, — the  limits  of  the  two  unequal 
bodies  may  be,  and,  in  fact  very  frequently  are,  incapable  of  co- 
incidence in  any  of  their  finite  parts.  In  order  to  make  surface 
a  mathematical  magnitude  (i.  e.,  definable  with  precision),  there 
must  be  found  at  least  one  homogeneous  surface,  of  which  any 
finite  part  is  capable  of  coincidence  with  any  other  correspond- 
ing part  of  the  same,  and,  after  taking  such  a  surface  as  stand- 
ard, there  must  be  found  rules  how  to  reduce  other  surfaces,  with 
any  desirable  degree  of  precision,  to  this  standard.  Such  homo- 
geneous surfaces,  the  essential  characteristic  of  which  is  that 
any  part  of  them  can  be  imagined  to  slide  upon  the  whole,  re- 
maining always  in  coincidence  in  all  its  parts  with  correspond- 
ing portions  of  the  whole,  do  really  exist ;  and  their  existence 
is  likewise  a  matter  both  of  experience  and  of  mathematical  de- 
duction. Any  of  these  homogeneous  surfaces  might  be  taken 
as  a  standard  of  measurement ;  but  one,  as  affording  the  greatest 
advantages  for  computation,  and  being  capable  of  indefinite  ex- 
tension, is  accepted  as  the  standard,  and  all  others  are  always 
reduced  to  this  single  standard.  (It  is  needless  to  remark  that 
a  difference  of  choice  of  the  standard  surface  would,  like  dif- 
ferent systems  of  numeration,  lead  only  to  different  methods  of 
computation,  but  not  to  different  results.) 

To  sum  up  : — surfaces  are  multiform  and  are,  therefore,  seldom 
capable  of  coincidence  in  their  finite  portions.  The  nature  of 
measurement,  however,  requires  a  homogeneous  standard,  to 
which  all  magnitudes  of  the  same  kind  that  are  to  be  meas- 
ured, can  easily  be  reduced.  Surfaces  answering  this  descrip- 
tion of  homogeneity  and,  hence,  capable  of  serving  as  a  standard 
of  measurement  of  area  and  form,  actually  exist  —  of  different 
species  and  infinite  in  number — the  essential  characteristic  of 
all  of  which  is  the  capability  of  any  portion  of  such  a  surface 
to  slide  along  the  whole,  remaining  always  in  coincidence  with 
different  corresponding  portions  of  the  whole.  The  simplest 
of  these  is  chosen  as  the  norm ;  it  will  be  shown  that  it  pos- 
sesses the  additional  properties  of  being  indefinitely  extended, 
beyond  any  arbitrary  limit,  and  of  its  side  towards  the  interior 
of  the  body  which  is  limited  by  it,  fitting  upon  the  opposite 


48 

exposed    side,    so    that   the    two    can    be   made   to    coincide 
(plane). 

Definition  13.  —  If  two  bodies  are  in  partial  contact  of  their 
surfaces  with  each  other,  the  boundary  separating  the  part  of 
surface  in  contact  from  the  part  not  in  contact,  in  either  of  the 
bodies,  is  the  limit  of  either  the  covered  or  the  exposed  portion 
of  the  surface.  For  simplicity,  let  us  imagine  the  surfaces  to 
be  homogeneous.  Any  finite  portion  of  the  uncovered  surface 
can  be  imagined  to  be  in  contact  with  the  surface  of  some  third 
body  (see  Def.  11),  whose  form  at  a  finite  distance  is  immaterial, 
and  which  moves  upon  the  rest  of  the  uncovered  surface,  along 
any  path  in  it,  until  it  reaches  the  limit  of  the  covered  surface, 
where  it  is  checked  in  its  motion  by  the  rigidity  of  the  surface 
of  the  covering  body,  and  can  only  move  so,  that,  while  a  por- 
tion of  its  surface  touches  the  covered,  another  portion 
touches  the  covering,  body.  Motion  along  the  boundary 
separating  the  covered  from  the  uncovered  surface,  must 
still  be  possible  for  an  indefinitely  small  portion  of  a  finite 
body,  whose  contact  with  the  two  bodies,  in  the  exposed  por- 
tions of  their  respective  surfaces,  blends  along  the  boundary 
for  the  infinitesimal  element ;  for,  as  this  finite  body  can  be 
conceived  to  move  along  the  two,  remaining  always  in  partial 
contact  with  each,  the  infinitesimal  element,  touching  the  two 
simultaneously,  must,  of  necessity,  find  a  region  of  motion 
along  their  common  boundary.  This  boundary  will,  therefore, 
have  parts  of  its  own,  viz.,  the  specializations  of  position  of 
the  infinitesimal  touching  element  considered ;  but  these  parts 
will  not  be  of  the  same  kind  as  the  parts  of  a  surface  ;  neither 
can  the  whole  be  a  part  of  surface.  In  fact,  it  cannot  be  a 
part  of  the  covered  surface,  since  it  must  likewise  belong  to  the 
uncovered  surface ;  but,  however  small,  a  portion  of  the  cov- 
ered surface,  if  not  infinitesimal,  will  always  have  some  of  its 
parts  removed  from  the  exposed  region  by  intervening  parts  of 
the  covered  region.  Similarly,  it  cannot  be  a  part  of  the  ex- 
posed surface.  But  we  have  proved  that  it  must  have  parts  of 
its  own.  The  boundary  between  two  portions  of  surface  is, 
therefore,  a  new  magnitude ;  it  is  a  line,  and  its  parts  are  dif- 
ferent from  those  of  a  solid  or  a  surface,  but  like  them  expres- 
sible in  numbers  having  some  determinate  relation  to  the  num- 
bers expressing  the  magnitudes  of  volume  and  surface.  (The 
same  reasoning  holds  also  in  the  case  of  non-homogeneous  sur- 


49 

faces,  provided  a  certain  amount  of  plasticity  of  form  is  al- 
lowed to  the  touching  parts  of  the  moving  body). 

Scholium.  —  The  analysis  of  the  last  Definition  can  be  made 
more  concrete  by  the  following  r6sum6,  which  also  puts  its  re- 
sult in  a  somewhat  different  light : — 

When  three  bodies  touch  one  another  in  their  surfaces  and 
they  also  fit  one  another,  so  that  no  vacant  space  is  left  between 
the  parts  of  the  touchings,  the  same  part  of  surface  can  belong 
only  to  two  of  the  touching  bodies  at  once  ;  while  the  boundary 
separating  the  surface  belonging  to  any  pair  simultaneously, 
from  the  surface  belonging  to  either  of  the  pair  and  the  re- 
maining third  body,  belongs  to  all  three  bodies  simultaneously 
and,  hence,  is  not  a  surface,  but  a  line.  It  is  the  continuous 
boundary  of  rigidity  of  three  bodies  that  have  come  into  con- 
tact with  one  another,  and  any  indefinitely  small  part  of  it  can 
be  conceived  to  pass  to  the  position  of  any  other  of  the  same 
only  by  two  different  courses,  in  case  the  boundary  is  com- 
pleted and  the  line  returns  into  itself,  and  only  by  one  course, 
if  the  boundary  is  not  completed. 

Corollary.  —  From  this  follows  immediately  the  statement 
made  in  Definition  12,  that  an  infinitesimal  part  of  a  surface  can 
move  towards  another  infinitesimal  part  of  the  same  surface, 
by  a  continuous  infinity  of  different  paths  in  the  very  surface. 

The  idea  of  a  line  is  also  transferred  from  a  physical  sur- 
face upon  a  geometrical  one,  and  the  geometrical  line  may 
be  said  to  represent  the  geometrical  place  of  a  physical  line. 
Geometry  regards  it  as  a  separate  entity,  which  can  be  conceived 
to  move  about  in  space,  or  upon  a  suitable  surface,  without 
change  of  form  or  magnitude. 

Definition  14.  —  Any  body  may  be  regarded  as  divided  into 
two  definite  parts,  having  one  part  of  their  surfaces  —  namely, 
that  created  by  the  section — in  mutual  contact,  while  the  surface 
of  the  original  whole  body  is  now  also  cut  into  two  parts,  each 
belonging  to  one  of  the  two  bodies  now  taking  the  place  of  the 
original  one.  The  boundary  separating  the  common  surface 
from  the  distinct  parts  will,  according  to  Definition  13,  be  a  line 
on  the  surface  of  the  original  body, —  thus  being,  from  one  side 
at  least,  exposed  to  space.  Any  part  of  this  line  may  be  brought 
into  contact  with  some  other  body.  When  the  whole  line  is  in 
contact  with  other  bodies,  leaving  no  part  of  it  adjacent  to  va- 
cant space,   it  can    be   regarded  as  covered  by  another,  the 


50 

duplicate  of  the  first  in  form  and  magnitude,  which  is  traced 
upon  the  surfaces  of  the  touching  bodies, —  and  no  body,  be- 
sides, can  be  in  contact  with  the  line.  The  line  and  its  du- 
plicate are  then  said  to  coincide  —  coincidence  in  this  case  also 
being  a  proof  of  equality  in  both  form  and  magnitude.  If  we 
form,  in  a  similar  way,  the  duplicate  of  only  a  portion  of  the 
line,  this  duplicate  will  evidently  be  less  in  magnitude  than  the 
whole  line. 

Corollary.  —  Lines  coincide  with  one  another,  when  the  sur- 
faces limited  by  them  coincide,  since  none  of  the  lines  limiting 
these  surfaces  can  be  within  or  mthout  the  limited  surfaces. 
The  converse,  however,  is  not  necessarily  true  (except  in  the 
case  of  the  plane,  as  will  be  shown  later). 

Scholium.  —  Lines,  like  surfaces,  are  multiform;  but  there 
exist  also  homogeneous  lines,  of  which  any  part  is  capable  of  coin- 
cidence with  any  corresponding  part  of  the  same.  One  of  such 
homogeneous  lines,  capable  of  indefinite  extension  and  uniquely 
determined  by  any  two  of  its  elements,  is  accepted  as  the  stand- 
ard of  line-measurement  (straight  line). 

Definition  15.  —  When  the  surfaces  of  two  bodies  are  in  par- 
tial contact,  the  bounding  line  being,  in  its  turn,  brought  into 
partial  contact  with  a  corresponding  line  upon  the  surface  of  a 
third  body,  or,  in  other  words,  a  part  of  the  line  being  covered, — 
any  indefinitely  small  part  of  the  remaining  uncovered  line  may 
again  be  covered  with  a  corresponding  infinitesimal  line  upon 
the  surface  of  still  another  body  ;  then  the  parts  of  the  last  body, 
immediately  adjacent  to  the  line  in  question,  may  be  imagined  to 
move  along  the  uncovered  part  of  the  line,  only  by  two  opposite 
paths,  until  they  reach  the  limit  of  the  covered  part  of  the  line, 
where  further  motion  is  checked  altogether  by  the  rigidity  of 
the  surface  of  the  body  which  has  effected  the  first  partial 
covering  of  the  line.  This  limit,  therefore,  separating  the 
covered  part  of  the  line  from  the  uncovered,  has  no  parts  at  all, 
since  an  infinitesimal  element,  coming  up  to  it,  finds  no  exten- 
sion to  move  upon.  (There  would  be  no  advantage  in  this  case 
in  starting  with  a  finite  part  of  the  line  moving  upon  the  un- 
covered part,  by  allowing  plasticity  of  form  in  case  of  non- 
homogeneity,  since  —  a  line  being  a  region  of  motion,  only  for  an 
infinitesimal  part  of  a  body,  touching  two  surfaces  at  once  — 
the  motion  upon  it  must  be  a  filing  in,  or  successional  motion, 
all  along  the  line ;  that  is,  the  motion  of  a  row  of  individual 


51 

members,  where  there  is  a  unicursal  succession  of  each  mem- 
ber into  the  place  of  the  one  immediately  preceding,  going  on 
either  indefinitely,  or  returning  to  the  original  starting  place, 
—  the  members  being  the  infinitesimal  portions  of  the  line,  and 
all  of  them  belonging  to  the  same  series.)  The  proof  that  the 
limit  we  have  found  is  a  part  neither  of  the  covered,  nor  of 
the  uncovered,  portion  of  the  line,  is  perfectly  similar  to  the 
proofs  given  for  the  limits  of  a  solid  and  a  surface,  and  need 
not  be  repeated  again.  The  boundary  of  a  line  is  no  magnitude. 
It  is  called  a  point  in  geometry,  which  regards  only  its  position 
or  geometrical  place.  We  say  its  motion  generates  a  line, 
meaning  by  this  that  a  line  represents  a  field  of  motion  for  it, 
or,  otherwise,  the  path  of  a  moving  point.  It  is  regarded  in 
geometry  as  a  separate  entity,  which  can  move  about  in  space 
independently.  It  is  neither  homogeneous  nor  heterogeneous, 
since  it  has  no  parts.  The  geometrical  places  of  two  points 
always  coincide  with  each  other,  as  soon  as  they  are  brought 
into  the  same  position  in  space,  within  a  body,  upon  a  surface, 
or  a  line.     It  is,  therefore,  regarded  as  the  element  of  space. 

Corollary. — When  two  lines  coincide,  their  ends,  or  the  points 
limiting  them,  coincide  also,  i,  e.,  these  ends  have  the  same 
positions,  two  by  two. 

To  sum  up  what  has  been  stated  in  the  foregoing  definitions : — 

The  boundary  separating  impenetrable  substance  from  capa- 
cious space,  or  the  region  where  no  motion  is  possible,  from 
the  region  where  motion  is  wholly  unimpeded,  is  a  surface,  and 
admits  only  of  motion  in  contact  along  finite  regions,  with  the 
condition  of  plasticity  of  the  touching  surface  of  the  moving 
body  for  the  case  of  non-homogeneity. 

The  boundary  separating  uncovered  surface  from  covered 
surface,  or  the  region  where  motion  in  contact  is  possible,  from 
the  region  where  motion  in  contact  for  finite  bodies  is  also  im- 
possible, is  a  line,  and  belonging  neither  to  the  first  nor  to  the 
second,  it  admits  motion  in  contact  for  an  indefinitely  small 
portion  of  a  body. 

The  limit  separating  an  uncovered  portion  of  a  line  from  a 
covered  portion,  or  the  region  where  motion  in  contact  is  pos- 
sible for  an  infinitesimal  portion  of  a  body,  from  the  region  of 
absolute  exclusion  of  motion,  is  a  point,  which  is,  thus,  the 
position  of  an  infinitely  small  portion  of  a  body  at  rest ;  it  has, 
therefore,  no  dimensions  and  only  position. 


62 

Space,  in  its  totality,  being  the  repository  of  extended  sub- 
stance which  is  capable  of  motion  (change  of  position)  and 
endowed  with  the  properties  of  impenetrability,  rigidity,  and 
infinite  divisibility,  limiting  and  bounding  vacant  space  in  cer- 
tain definite  ways, — gives  rise  to  three  difiPerent  kinds  of  spacial 
magnitude,  so  connected  that  one  is  the  limit  of  the  other  and 
is  limited  by  the  third.  The  point  is  the  result  of  limitless 
divisibility  of  any  of  the  three  kinds  of  extended  spacial  mag- 
nitude. 

Space  is,  therefore,  a  tridimensional  manifoldness,  only  be- 
cause of  its  three  chief  attributes,  giving  three  different  kinds 
of  specializations  of  position,  limiting  each  other  and  so  con- 
nected that  there  is  always  a  certain  determinate  relation  be- 
tween the  units  of  one  kind  of  space  and  the  units  of  the  other 
kinds.  It  was  shown  in  Definition  11,  that  the  rigid  surface 
by  means  of  which  the  property  of  impenetrability  makes  itself 
effective,  must  be  some  function  of  the  volume ;  reasons  were 
also  given  why  a  surface  should  be  a  field  of  motion  for  finite  re- 
gions in  contact,  and  a  line,  a  field  of  motion  for  infinitesimal 
regions  of  contact ;  from  which  will  follow  at  once,  that  a  line 
is  a  differential  element  of  a  surface,  and  a  point,  an  element  of 
a  line.  Limitless  space,  in  its  totality,  would  be  a  one-dimen- 
sional magnitude,  ranging  from  zero  to  infinity,  in  terms  of 
volume  alone, — were  it  not  for  the  invariable  relations  between 
the  units  of  volume  and  those  of  the  two  subcategories  of  speci- 
fied space,  resulting  from  the  fact  that  they  always  limit  one 
another.  In  order,  therefore,  to  suppose  that  space  has  more 
than  three  dimensions,  we  must  conceive  that  a  point  has  di- 
mensions, since  we  began  our  analysis  from  free  unlimited  space, 
and  found  it,  in  itself,  without  considering  its  limits,  to  be  one- 
dimensional  ;  and  only  in  relation  to  its  limits  does  it  become 
tridimensional,  since  its  first  two  limits  (but  not  the  third)  are 
also  magnitudes  and  bear  certain  fixed  relations  to  unspecified 
space  as  a  magnitude.  Since  u,  du,  d^u^  where  u  represents  a 
piece  of  unspecified  space,  i.  e,  volume,  are  all  variables,  but 
not  so  d^Uj  it  follows  that,  if  u  is  a  function  of  .r,  it  must  be  of 
the  third  order  :  lo  =  ax^  may  do  for  the  simplest  representation 
of  such  a  function,  where  dx  will  represent  the  differential  of 
a  line  ;  dx  may,  of  course,  be  a  homogeneous  function  of  the  first 
degree,  of  a  number  of  differentials  dx^,  dx^,  dx^,  •  •  ■  dx^ ;  but 
then  there  must  be  ti  —  3  linear  relations  between  the  dx^a,  re- 


53 

ducing  the  number  of  independent  ones  to  three.  It  is  absurd, 
by  reasoning  in  a  reversed  order,  to  infer  by  a  kind  of  induc- 
tion, that,  just  as  a  point  in  moving  generates  a  line,  a  line, 
in  moving  out  of  its  regions,  generates  a  surface,  and  a  sur- 
face generates  a  body,  so  tridimensional  space,  in  moving 
out  of  itself,  will  produce  a  new  kind  of  space.  In  the  first 
'place,  we  would  have  to  prove  in  general,  that  if  our  reasoning 
holds  for  n,  it  will  hold  for  n  +  1,  as  we  always  do  in  mathe- 
matics in  such  a  kind  of  induction.  Riemann^s  construction  of  an 
(n  -f-  l)-fold  variability,  out  of  an  n-fold  one  and  a  variability 
of  one  dimension,  is  based  on  the  assumption  that  the  n-fold 
variability  passes  over  into  another  one,  entirely  differe'iit,  in  a 
determinate  way,  so  that  each  point  of  the  first  passes  over  into 
a  definite  point  of  the  other,  which  is  not  at  the  same  time  a  ptolnt  of 
the  first,  —  an  assumption  that  must  he  proved  in  each  particular 
case.  In  our  case,  this  assumption  is  actually  equivalent  to  as- 
suming, that  space  can  move  out  of  space,  which  is  absurd  by  the 
very  definition  of  space,  viz.,  space  is  that  which  gives  place  to 
material  objects,  whether  at  rest  or  in  motion  ;  so  that  wherever 
motion  is  at  all  possible  for  a  tri-dimensional  piece  of  space,  there 
is  space  again.  In  the  second  place,  even  if  we  admit  the  possi- 
bility of  space  moving  out  of  itself  into  some  other  region,  we  have 
not  admitted  any  new  property  of  space  which  might  be  the  ob- 
ject of  measurement,  since  we  tacitly  assume  that  the  new  region 
is  not  space,  so  that  space  would  remain  again  tridimensional. 
And  even  if  we  should  admit  a  new  unknown  property  of 
space,  we  would  still  have  to  prove  that  tridimensional  space 
is  its  limit ;  and  that  its  units  bear  a  fixed  mathematical  rela- 
tion to  the  units  of  space  we  have  already  considered.  Indeed, 
it  is  not  sufficient  for  a  phenomenon  to  have  a  certain  number  of 
properties,  in  order  to  consider  that  phenomenon  of  as  many 
dimensions  as  the  number  of  its  properties.  The  properties 
must  be  the  limits  of  each  other,  and  their  units  must  stand  in 
a  certain  invariable  mathematical  relation  to  one  another. 

The  following  is  an  analytical  deduction  of  the  number  of 
dimensions  of  space  considered  as  a  point-manifold,  which 
was  written  up  and  added,  as  a  supplement  to  the  introductory 
chapter  of  the  main  body  of  the  Dissertation,  two  years  after 
the  latter  had  been  completed.  The  purpose  of  this  analysis 
is  to  prove  the  a-priori  necessity  of  three  dimensions,  when  the 
point,  as  usually  defined  in  elementary  geometry,  is  taken  as 


54 

the  element  of  space.  It  will  follow  that  to  make  space  a  four- 
dimensional  manifold,  without  changing  its  element  to  some 
other  geometrical  entity,  will  involve  a  contradiction  in  terms. 
The  discussion  is  divided  into  13  paragraphs.  — 

1)  Let  the  whole  original  manifold  he  S,  which  we  suppose  to 
be  continuous  J  i.  e.,  any  two  different  positions  in  it  can  be 
reached  from  each  other  only  through  an  unbroken  series  of  other 
positions,  all  in  S,  whose  number  is  infinite. 

2)  Let  an  invariable  piece  of  it  —  11,  endowed  with  impen- 
etrability, rigidity,  and  infinite  divisibility,  be  imagined  as 
capable,  as  a  whole,  of  changing  its  position  in  S.  The  invari- 
ability of  U  is  characterized  by  the  fact  that  the  mutual  dispo- 
sition, or  arrangement,  and  the  relation,  of  the  parts  of  f/,  ob- 
tained in  a  determinate  way  by  any  arbitrary  subdivision,  is  to 
remain  unaltered  with  respect  to  one  another  and  with  respect 
to  the  whole  of  U,  considered  as  an  entire  manifold  in  itself. 
We  say  that  internal  motion,  of  parts  of  U  within  C7,  is  ex- 
cluded, and  any  two  parts  that  have  been  separated  from  each 
other  by  certain  continuous  series  of  other  parts,  in  one  posi- 
tion of  U  within  >S^,  will  remain  so  in  any  other  position.  We 
thus  arrive  at  the  notion  of  Bulk  or  Volume.  Further,  this 
notion  is  made  more  precise  by  postulating,  that,  when  such  a 
rigid  piece  has  once  occupied  a  definite  portion  of  vacant  or 
unoccupied  8,  to  the  exclusion  of  any  other  impenetrable 
piece,  it  will  always  be  brought  into  coincidence  with  that 
portion  again,  as  soon  as  a  finite  part  of  it,  no  matter  how 
small,  will  be  brought  into  coincidence  with  the  corresponding 
part  of  8  it  has  occupied  originally  ;  any  other  small  portion 
of  U  will  then  also  have  to  occupy  its  original  position  within 
8.  (According  to  this  postulate,  it  is  perfectly  indifferent 
whether  Z7is  conceived  to  move  within  8,  or  U  is  conceived 
as  stationary  and  8  as  changing  its  relation  with  respect  to 
Uf  in  giving  it  position  in  different  portions  of  itself — all 
these  portions  being,  of  necessity,  equal  in  bulk  or  volume  to 
one  another  and  to  U.)  If  we  should  measure  only  bulk  or 
volume,  we  would  get  only  one  dimension.  The  property  of 
impenetrability  of  the  movable  pieces,  however,  leads  us  to 
distinguish  a  new  category  of  manifold,  subordinate  to  the  cate- 
gory >S'  and  contained  in  it,  in  the  following  way  :  — 

3)  Suppose  U  fixed,  and  another  piece  F,  of  same  nature, 
moving  up  to  it,  reaches  the  boundary  of  the  latter.     It  will 


65 

then  be  prevented  from  occupying  the  same  place  as  U  by  the 
impenetrability  of  the  two,  exhibited  in  their  boundaries,  so 
that  a  certain  kind  of  motion  of  V  is  checked  when  the  latter 
comes  into  contact  with  U.  Here  we  have  the  notion  of  the 
boundary  of  U  separating  it  from  vacant  Sy  or  from  other  pieces 
V  of  same  nature  as  itself. 

4)  This  first  derivative  boundary  of  U,  which  we  may  denote 
by  Z7',  is  neither  a  portion  of  Z7,  nor  of  V  that  has  come  into 
contact  with  Uj  nor  of  the  vacant  8,  in  which  U  is  posited ;  it 
may,  or  may  not,  have  portions  of  its  own.  Example  : — the 
limits  of  a  period  of  time,  no  matter  how  great,  have  no  parts, 
since  there  is  no  possibility  for  an  invariable  piece  of  time  to 
change  its  position  with  respect  to  other  invariable  pieces,  and 
move  up  to  them  to  come  in  contact  with  their  boundaries.  In 
our  case,  however,  U'  must  have  parts  of  its  own,  as  will 
become  evident  from  the  following  considerations  : 

5)  By  infinite  divisibility  of  a  rigid  piece  F,  we  may  arrive 
at  the  notion  of  a  plastic  substance,  each  infinitesimal  portion 
of  which  occupies  a  corresponding  infinitesimal  portion  of  the 
manifold  S,  and  the  whole  retaining  only  the  property  of  im- 
penetrability, having  lost,  however,  rigidity.  It  will  repre- 
sent, in  fact,  a  plastic  substance,  the  smallest  portions  of  which 
are  easily  capable  of  separation  and  change  of  position  with 
respect  to  one  another  and  with  respect  to  the  whole  of  their 
aggregate.  The  rigid  piece  U  considered,  if  immersed  in  this 
plastic  substance,  will  displace  a  portion  of  it  equivalent  to  the 
bulk  of  the  portion  immersed.  To  a  smaller  portion  of  U  im- 
mersed will  correspond  a  smaller  bulk  of  the  plastic  material 
displaced,  and  to  a  greater  portion  immersed  will  correspond  a 
greater  bulk  displaced.  Now  this  displacement  is  necessarily 
effected  only  by  the  boundary  of  the  immersed  portion  of  Uj 
and  by  that  part  of  it  alone  which  has,  before  immersion,  been 
exposed  to  vacant  /S',  as  distinguished  from  the  remaining  part 
which  —  separating  the  immersed  portion  of  U  from  the  non- 
immersed — could  have  no  effect  in  the  act  of  displacement 
considered.  It  follows,  therefore,  that  to  a  greater  bulk  dis- 
placed— and  hence  to  a  greater  portion  of  U  immersed — cor- 
responds a  greater  portion  of  the  boundary  U' y  which  we  may 
call  C7.'  (i  =  immersed),  as  distinguished  from  the  remaining 
portion  of  U'y  which,  as  belonging  to  the  exposed  portion  of 
U  only,  we  may  call    Z7J  (e  =  exposed).     By  changing  con- 


56 

tinuously  the  portion  of  U  immersed,  from  an  infinitesimal 
bulk  to  the  whole  bulk  of  U,  we  arrive  at  the  notion  of  a  con- 
tinuously increasing  boundary  1[J[,  from  an  infinitesimal  to  the 
whole  of  U' ,  and  a  correspondingly  decreasing  Z7J,  from  the 
whole  of  U'  to  an  infinitesimal  of  f/J,  and  then  to  zero. 

6)  The  limit  separating  the  two  portions  of  V  (i.  e.j  U[  from 
CT),  at  each  and  every  stage  of  the  process,  is  evidently  a  new 
kind  of  boundary — :  TJ",  —  and  the  infinitesimal  portion  of 
U\  between  two  very  near  Z7"'s  corresponding  to  two  very 

nearly  equal  bulks  of  TJ  immersed,  in  the  continuous  process 
described  above,  may  be  denoted  by  dU' ,  meaning  an  infini- 
tesimal of  TJ' y  and  the  aggregate  of  all  these,  ^dUlj  may  be 
taken  to  equal  the  whole  of  U'  belonging  to  the  whole  of  U, 
just  as  ?7=  ^dUj^.  (In  the  synthetic  discussion  of  the  dimen- 
sions, above  (Definition  12),  allusion  was  made  to  the  geomet- 
rically indispensable  notion  of  a  homogeneous  Z7',  which  might 
serve  as  a  standard  of  measurement  for  different  C/''s,  and 
later  the  actual  existence  and  construction  of  such  a  U'  will 
be  rigorously  proved.)  A  homogeneous  U'  may  be  considered 
a  field  of  motion  for  finite  portions  of  C7',  covered  by  corre- 
sponding portions  of  V  belonging  to  some  moving  rigid  F, 
which,  in  the  process  of  motion,  touches  U  in  the  variable  por- 
tions of  U'  considered.  U'  in  itself,  without  the  piece  U 
which  is  bounded  by  it,  has  an  independent  existence,  only  as  an 
abstraxition  (like,  for  instance,  force  without  matter).  In  fact, 
we  can  speak  of  it  as  moving  about,  either  in  8  or  in  its  own 
region,  only  in  vii^tue  of  corresponding  motions  of  U  (or  V)  to 
which  U'  (or  V)  belongs. 

7)  Corollary.  —  It  follows  also,  that  U',  as  a  boundary  be- 
tween U  and  unoccupied  adjoining  portions  of  S,  or  between 
U  and  F,  must  be  considered  as  having  the  property  of  im- 
penetrability. In  fact,  U'  is  conceived  of  as  that  which  pre- 
vents any  portion  of  F,  no  matter  how  small,  from  penetrating 
into  the  region  occupied  by  any  portion  of  Z7,  and  vice  versa. 
But,  as  a  region  of  motion  for  portions  of  itself  i.  e.,  as  a  mani- 
fold in  itself  it  cannot  possess  the  property  of  impenetrability,  and 
must,  in  fact,  be  of  the  same  nature  with  respect  to  portions  of  V 
or  U'  moving  in  the  whole,  as  vacant  S  is  with  respect  to  impene- 
trable U  or  F.  This  is  a  reason  why  the  boundary  between 
U^  and  Cr  can  be  obtained  only  through  the  medium  of  an 
auxiliary  F,  which  may  also  be  considered  rigid,  and  portions 


57 

of  whose  boundary  —  F',  very  near  F'(  =  U^y  c  =  covered), 
will  then  serve  as  a  check  to  a  third  rigid  piece  Wy  of  same 
nature  as  U  and  V,  a  finite  portion  of  whose  boundary  —  W^ 
is  conceived  as  covering  corresponding  finite  portions  of  UJ 
(exposed  with  respect  to  V  only)  and  moving  in  the  manifold 
Z7',  until  a  finite  portion  of  W^  comes  into  coincidence  with  a 
corresponding  portion  of  F',  which  thus  becomes  V^^  =  W^^. 

8)  It  is  important  to  observe  that,  when  a  piece  of  the  original 
manifold  F  comes  into  coincidence  with  another  piece  of  the 
same  kind  U",  in  general  only  finite  portions  of  their  boundaries 
coincide  and  thus  become  IT  =  FJ  ;  the  remaining  portions  of 
their  boundaries  combine  in  forming  a  new  combined  boundary 
of  the  piece  {U-j-  F),  taken  as  a  whole,  —  this  combined 
boundary  being  now  represented  by  (^7+ F)'=  C/"^' -f- Fj'.* 
It  is,  therefore,  evident  that,  when  TF,  considered  in  No.  7, 
moves  up  to  F,  the  two  portions  of  its  boundary  —  W^^  and 
W^^ — become  now  combined  into  the  boundary  separating  W 
from  the  combined  piece  {U-i-  F),  m., 

(wi  +  w:j={u+v):j 

and,  in  general,  for  the  same  reason  as  above,  there  will  yet  re- 
main a  finite  portion  of  ( Z7  +  F)'  exposed  which  we  may  denote 
hyiU+V):. 

9)  We  see  now  that  U'\  —  originally  obtained  as  the  limit 
separating  U!  from  C7',  when  F  was  considered  as  the  plastic 
substance  in  which  a  portion  of  U  was  immersed,  and  then 
identified  with  the  boundary  separating  U^{—  FJ)  from  C/',  and 
also  from  F',  in  case  Fis  considered  rigid  and  a  portion  of  its 
boundary  FJ  covering  an  equal  portion  of  C/', —  by  means  of 
the  process  considered  in  No.  7  becomes  broken  up  into  two  por- 
tions :  one  lying  in  the  region  of(U+  F)^  exclusively,  and  sepa- 
rating U^^  from  Fj"  ,  or  each  of  these  from  the  same  correspond- 
ing portion  of  U^(=  F^'),  and  the  other  lying  exclusively  in 
the  region  of  (C/-h  F)f^  =  (TF,;  +  TF/J,  and  separating  U;^ 
from  V'  .  Let  the  first  be  called  U''  and  the  second  U'J . 

10)  By  introducing  another  piece  T,  which  we  make  to  play 
the  r6le  of    W  for  the  breaking  up  of   U^l  into    C/^J^  ^^  and 

*  This  combined  boundary  is  in  no  way  different  in  character  from  U', 
being  like  it  continuous,  and  having  finite  parts  represented  by  A  ( C7"+  T)' 

-=AC7^  +  AFe'. 


58 

U"  c ,  and  then  still  another  piece  X  for  the  breaking 
up  of  U''  ^^  into  two  pieces,  and  so  on,  we  see  that  U"  consists 
of  as  many  parts  as  we  please.  Moreover,  by  making  W  move 
lip  to  T,  so  that  (  W'^^  -f  M7J,  —  conceived  as  changing  its  posi- 
tion continuously  and  as  changing  its  form  and  magnitude  if 
need  be,*  —  shall  always  remain  during  this  process  in  coinci- 
dence with  an  equivalent  variable  (  C/ +  F)^^,  we  shall  convince 
ourselves  that  an  infinitesimal  d{W^^-\-  TF^'J,  in  the  neigh- 
borhood of  U",  remaining  always  in  coincidence  with  a  variable 
d{TJ-\-V)'^^,  Avill  displace  itself  and  find  a  region  of  motion 
along  the  element  d{U -\-  V)'  taken  in  the  neighborhood  of  U" 
and  conceived,  in  toto,  as  a  locus  for  the  different  positions  of  the 
moving  infinitesimal  portion  of  the  first  derivative  boundary. 
U"  itself,  therefore,  as  a  whole,  will  prove  to  be  a  locus  in  quo 
for  U'J^  —  Urn  d{U  +  V)l^.     When  W comes  up  to  T,  it  is  there 

checked,  —  and  we  arrive  at  the  conception  of  U'",  separating  the 
region  of  motion  for  U^'^  from  the  region  where  such  a  motion 
is  impossible.  It  is,  of  course,  the  same  boundary  as  that 
which  separated  U^'^^^t  fronn  L^^^  ^^,  obtained  at  the  beginning  of 
this  paragraph. 

11)  In  summing  up,  we  see  that,  while  S  is  a  region  of  mo- 
tion for  pieces  like  C/,  F,  etc.,  the  first  derivative  boundary 
U'  is  the  limit  of  d  U,  and  that,  not  being  a  portion  of  U,  it  has 
portions  of  its  own,  being  a  region  of  motion  for  corresponding 
'finite  portions  of  the  boundaries  of  a  movable  covering  piece 
F.      IJ"  is  the  limit  of  dU'  as  dU'  =  0,  and,  not  being  a  part 
of  Z7',  on  account  of  its  sej^arating  the  region  of  motion  in  con- 
tact for  a  finite  piece  W  from  the  region  where  such  a  motion 
is  impossible,  it  still  has  portions  of  its  own.     For,  although 
W  cannot  find  upon  it  a  region  of  motion,  even  for  a  finite 
portion  of  its  boundary,  it  can  find  upon  it  a  region  of  motion 
for  the  limit  of  an  infinitesimal  portion  of  its  boundary,  namely 
for  lim  dW  (=  W").     For,  as  it  was  shown,  TFcan  be  made 

0 

to  move  so,  that,  while  two  finite  portions  of  its  boundary,  dis- 
tinct but  contiguous  in  W",  move  each  upon  a  corresponding 
portion  of  U'  and  F'  respectively,  an  infinitesimal  of  its  boun- 
dary d  W,  contiguous  to  TF",  and  taken  as  near  W"  as  we  please, 

*  This  was  previously  shown  to  be  possible,  by  supposing  the  first  deriva- 
tive boundary  to  be  homogeneous,  for  simplicity,  or  by  allowing  sufficient 
plasticity  to  W\ 


69 

on  either  side  of  it,  will  move  and  remain  in  coincidence  with 
corresponding  variable  infinitesimal  portions  of  both  U'  and 
V ,  In  passing  as  near  to  the  limit  as  we  please,  we  come  to 
the  conception  of  portions  of  the  limit  W"  itself,  moving  in  the 
whole  of  it  as  in  a  locus  in  quo. 

The  third  derivative  boundary  U"  separates  the  region  of 
motion  for  the  limit  of  an  indefinitely  small  portion  of  the 
boundary  of  the  first  order  from  the  region  where  motion  is 
impossible  even  for  the  limit  of  an  infinitesimal  portion  of  such 
a  boundary.  An  infinitesimal  portion  of  U",  very  near  U'",  on 
both  sides  of  it,  namely  dU'\  may  be  considered  an  element 
of  U"j    so  that  we  have  ^d  U"  =  U", 

12)  And  now,  if  we  agree  that  any  manifold  derived  from  8 
in  the  manner  indicated,  will  have  to  be  capable  of  giving  place 
to  impenetrable  substance,  or  to  its  boundary  which  is  directly 
connected  with  the  substance,  we  find  that  the  last  derived 
boundary,  TJ"\  which  does  not,  even  near  its  limit,  afford  a 
region  of  motion  to  an  infinitesimal  of  the  first  derivative 
boundary  d  U'  of  the  impenetrable  substance,  has  only  position 
but  no  dimensions.  So  that  we  have  three  categories  of  space, 
corresponding  to  three  properties  of  bodies  —  rigidity,  im- 
penetrability, and  infinite  divisibility,  —  each  limiting  the  pre- 
ceding and  limited  by  the  following. 

These  are  :  — 

8,  a  region  of  motion  for  pieces  of  the  impenetrable  substance 
itself —  Uy  F,  etc.,  and  their  boundaries,  both  finite  and  infini- 
tesimal ; 

U\  the  first  derivative  boundary  of  impenetrable  substance, 
a  region  of  motion  in  contact  of  finite  pieces  like  Z7,  F,  i.  e.,  a 
region  of  motion  for  a  finite  portion  of  the  boundary  of  a  piece 
of  impenetrable  substance,  and,  lastly, 

Z7",  the  second  derivative  boundary,  a  region  of  motion  for 
the  limit  of  an  infinitesimal  portion  of  the  first  detnvoiive  boun- 
dary dU\ 

The  boundary,  U'",  between  two  portions  of  this  last  region 
of  motion,  because  of  its  limiting  the  region  of  motion  for  an 
infinitesimal  portion  of  the  boundary  belonging  to  impenetrable 
substance,  from  the  region  of  no  motion  even  for  an  infinitesi- 
mal portion  of  the  boundary,  must  be  of  zero  dimensions,  but 
still  capable  of  having  definite  position,  being  the  primary  irre- 
ducible element  of  space.     It  follows,  therefore,  since  each  of 


60 

the  boundaries  is  capable,  near  the  corresponding  limit,  of 
being  considered  an  infinitesimal  element  of  the  category  which 
is  bounded  by  it,  the  last  category  in  order  derived,  namely 
U"y  has  one  dimension  ;  the  one  preceding,  two  dimensions  ;  and 
the  original  one  U,  or  a  piece  of  vacant  S  as  measured  by  C7, 
three  dimensions. 

13)  *  By  the  very  process  of  the  deduction  of  the  three 
qualitatively  different  categories  of  space  (regions  of  different 
kinds  of  motion,  which,  however,  are  the  boundaries  of  each 
other,  in  a  series),  we  have  arrived  at  the  notion  of  a  manifold 
of  two  dimensions,  objectively  not  independent  from  the  main 
category,  but,  none  the  less,  having  a  true  abstract  reality,  and 
which,  by  its  very  nature,  as  a  manifold  in  itself  and  not 
as  a  boundary,  is  devoid  of  the  property  of  impenetrability 
(see  No.  7).  In  this  derived  manifold,  therefore,  boundaries 
of  portions  can  be  established  only  by  means  of  a  piece  of  the 
higher  manifold,  having  the  property  of  impenetrability  in  the 
region  of  the  dimensions  of  the  derived  manifold  considered, 
and,  for  this  very  reason,  suggesting  a  dimension  over  and  above 
those  of  the  derived  manifold.  It  is  no  wonder,  then,  that  the 
general  reasoning,  applicable  to  the  original  manifold,  is  not 
applicable  to  the  derived  manifold.  The  figure  given  as  an 
objection,  instead  of  disproving  the  reasoning,  is  only  another 
proof  of  its  validity.  Mark,  that  in  order  to  attribute  impene- 
trability to  the  limits  of  the  circle,  you  must  postulate  it  to 
be  infinitely  thin,  and  V  an  infinitely  thin  film,  —  which,  of 
course,  is  equivalent  to  postulating  a  third  dimension,  but  in  a 
very  disguised  form. 

Now,  this  infinite  thinness  is  already  capable  of  being  increased 
indefinitely.  In  other  words,  the  assumption  of  impenetrability, 
by  the  reasoning  employed  above,  would  involve  a  third  dimen- 
sion, outside  of  the  given  manifold  of  two  dimensions,  leaving 
this  last  unchanged.  A  reference  to  Nos.  7,  8,  9,  10  will  fully 
justify  this  assertion. 

And  still  otherwise,  —  perhaps  this  way  of  looking  at  the 
thing  may  be  more  satisfactory  :  —  To  such  intelligent  beings 

*  13)  is  a  reply  to  an  objection  raised  during  a  conference  -when  this  was 
presented  :  Why  by  the  same  general  reasoning  we  are  unable  to  prove  a 
third  dimension  even  in  the  domain  of  an  admittedly  two  dimensional  mani- 
fold, as  is,  for  instance,  the  first  derivative  manifold.  The  nature  of  the  ob- 
jection will  be  understood  by  the  reader  from  the  reply,  which  alone  is  given 
here  explicitly. 


61 


as  would  have  no  sense  of  the  third  dimension,  if  such  beings 
were  at  all  possible,  the  speculation  about  a  third  dimension 
would  not  only  involve  no  logical  contradiction,  but,  on  the  con- 
trary, would  be  a  perfectly  logical  and  necessary  generalization. 
For,  they  would  have  to  postulate  some  dimension — not  directly 
given  in  experience  —  as  a  medium  for  the  continuous  passage 
of  W  to  Ty  remaining  always  in  contact  with  both  U  and  F, 
which,  even  in  two  dimensions  are  not  essentially  separated, 
since  they  are  both  on  one  and  the  same  side  of  U'  and  V, 
But  the  speculation  about  a  fourth  dimension  for  such  beings  as 


a 

S 

.§ 

T 

'O 

*6 

^ 

"^ 

V 

O     ^ 

y' 

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-^    "fi            / 

f 

1  \ 

o  S       / 

j       \ 

*»   «      / 

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M          / 

\ 

(U    o 

\ 

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1.1 

S    o      \ 

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V" 

i  i 

^■s    \ 

Ci       OB 

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i  M 

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•^     bo 

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^^ 

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t^-^ 

c^ ■ 

f^ 

k5^1 

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Fig.  a. 


have  already  risen  to  the  empirical  verification  of  the  abstract 
deduction  of  three  dimensions,  would  certainly  involve  a  logi- 
cal inconsistency.  For,  starting  with  the  manifold  S  as  de- 
fined above,  in  its  most  general  aspect,  without  boundaries  at 
all,  and,  for  all  we  know,  having  n  dimensions,  where  n  may 
be  any  entire  positive  number,  we  were  led,  by  a  simple  analysis 
of  its  definition,  to  three  different  manifolds,  containing  each 
other  in  series,  and  the  third  derivative  boundary  which  is 
the  boundary  of  the  lowest  category  in  this  series,  was  found 
to  be  something  that  can  have,  at  most,  position,  but  no  dimen- 
sions. If  preferred,  you  may  say  that  we  have  proved  that  the 
maximum  number  of  dimensions  of  space  as  defined  by  the 
properties  of  8,  is  three,  and  that,  having  logically  arrived  at 


62 

the  maximum,  we  find  it  in  perfect  agreement  with  our  intuitive 
experience,  which,  of  course,  also  served  us  as  a  starting  basis 
in  defining  the  manifold  S,  at  the  beginning  of  the  present  dis- 
cussion. 

I  deem  it  necessary  to  repeat  at  the  conclusion  that  I  ac- 
knowledge the  fruitfulness  of  the  idea  of  making  space  a  mani- 
fold of  a  higher  number  of  dimensions,  by  dropping  the  property 
of  impenetrability  in  the  physical  sense,  and  assuming  a  figure 
depending  on  n  parameters,  as  the  element  of  space. 


Chaptek  II. 

THE  SPHERE,    THE  CIRCLE,    THE    STRAIGHT    LINE,    THE  ANGLE,  THE 
TRIANGLE,  THE  PLANE,  ETC. 


Definition  I.  —  A  pair  of  fixed  points  in  space,  or  any  two 
points  in  rigid  connection  which  can  he  made  to  coincide  with  these, 
are  said  to  he  at  an  invayiahle  distance  from  each  other. 

Corollary  I.  —  If  a  pair  of  points,  A  and  B,  in  rigid  con- 
nection, are  capahle  of  coincidence,  one  hy  one,  with  another  pair, 
C  and  D,  likewise  in  rigid  connection  between  themselves,  the  two 
pairs  loill  he  at  equal  distances,  each  point  from  its  pair,  i.  e., 
distance  of  B  from  A  =  distance  of  C from  D  or  of  D  from  C. 

Corollary  II.  —  Two  pairs  of  fixed  points  in  space.  A,  B,  and 
C,  D,  hoth  of  which  are  capahle  of  coincidence  successively  with 
the  same  freely  movahle  pair  in  rigid  connection,  E  and  F,  are  re- 
spectively at  equal  distances. 

For  shortness  we  shall  call  a  pair  of  points  in  rigid  con- 
nection, free  to  move  as  a  whole,  simply  a  rigid  pair  of  points, 
and  will  denote  them  thus  (AB). 

Axiom  1.  —  If  any  surface  or  line  can  be  made  to  coincide 
with  another  surface  or  line,  it  can  do  so  only  by  passing  from 
one  position  to  the  other  by  a  continuous  path,  consisting  of  an 
infinity  of  such  positions,  every  position  in  which  is  a  surface  or 
line,  respectively  congruent  with  the  moving  one. 

Lemma  1.  —  From  the  principle  of  rigidity  and  the  defini- 
tions of  body,  surface,  line,  and  point,  it  follows  that  a  body  is 
absolutely  fixed  in  space  when,  and  only  when,  a  finite  portion 
of  the  surface  limiting  it,  or  in  any  way  rigidly  fixed  in  it,  is 
fixed  in  space. 

For,  whenever  a  body  is  fixed  in  space,  the  whole  of  its  sur- 
face, limiting  it  from  all  space  around,  and  hence  every  finite 
portion  of  this  surface,  is  fixed  in  space ;  and  whenever  the 
body  is  moved,  the  whole  of  its  surface,  and  hence  every  finite 
portion  of  it,  no  matter  how  small,  changes  its  position.  We 
cannot  say,  however,  the  same  of  a  point  in  the  surface,  since 
a  point,  having  no  magnitude,  but  only  position,  does  not  limit 
the  surface  to  which  it  belongs  and  which  may  be  conceived  to 

63 


64 

change  its  position  as  a  whole,  and  hence  to  interchange  the 
positions  of  all  congruent  lines  in  it  that  are  drawn  from  a 
common  point  limiting  them  all  and  remaining  fixed.  Hence, 
a  body  and,  therefore,  also  any  surface  or  line,  rigidly  fixed  in 
it,  may  be  conceived  to  move  when  only  one  point  in  the  body, 
the  surface,  or  the  line,  is  held  fixed  in  space.  Such  a  motion 
of  the  body  is  called  rotation,  revolution,  or  turning,  about  the 
fixed  point. 

Theorem  1.  —  If  (AB)  denote  a  rigid  pair  of  points,  of  which 
A  is  fixed  and  (B)  is  made  to  assume  all  possible  positions 
compatible  with  the  rigidity  of  the  pair  and  the  fixity  of  A, 
then  (B)  will  describe  a  homogeneous  surface  called  a  spherical 
suifcvce,  limiting  a  body  called  a  sphere,  A  is  called  the  center  of 
the  sphere  J  or  of  the  spherical  surface,  and  the  invariable  distance 
AB,  from  the  center  to  any  point  in  the  surface,  is  called  the 
radius  of  the  sphere. 

In  fact,  the  moving  point  (B)  can  pass  from  any  fixed  point 
B  with  which  it  initially  coincided  to  any  other  point  J5',  J5", 
and  so  on,  by  a  continuous  infinity  of  paths  crossing  and  re- 
crossing  one  another  in  all  conceivable  ways,  provided  all  these 
points  are  at  the  same  distance  from  A  sls  B  (preceding  Lemma 
and  Axiom  1).  Let  (B)  pass  from  B  to  B'  by  a  continuous 
path  of  some  determinate  rigid  form,  BB';  while  doing  so,  any 
rigid  line  connecting  A  and  (jB)  will  describe  a  portion  of  a 
surface.  This  surface,  in  its  turn,  conceived  as  a  rigid  form, 
can  (according  to  Lemma)  be  imagined  so  to  move  while  A  is 
fixed,  that  every  one  of  its  points  describe  a  line  not  already  con- 
tained in  the  original  position  of  the  surface  itself, — thus  describ- 
ing a  body.  The  line  (BB'),  conceived  rigid  and  always  limit- 
ing the  moving  surface  {ABB'),  will  then  be  dragged  along  with 
it  in  its  motion  and  will  describe  a  surface,  since  each  of  its 
points  describes  a  line  not  coincident  with  the  original  position 
of  the  moving  line  ;  in  other  words,  each  point  of  the  line 
(BB')  passes  to  another  point  not  already  contained  in  the 
original  position  of  the  line.  Moreover,  since  the  surface 
{ABB')  can  be  conceived  to  sweep  through  all  that  portion  of 
space  around  A  whose  aggregate  of  points  is  characterized  by 
the  property  that  their  distances  from  A  are  correspondingly 
equal  to  those  of  the  aggregate  of  points  contained  in  {ABB') 
in  its  original  position,  it  follows  that  the  surface  described  by 
the  line  {BB')  will  contain  the  whole  aggregate  of  points  which 


65 

are  at  the  same  distances  from  A  as  (B)  in  its  original  posi- 
tion. It  is  also  evident  from  the  moae  the  body  described 
by  the  surface  (ABB')  is  generated,  that  it  is  a  continuous 
body,  i.  e.y  the  aggregate  of  points  composing  it  is  a  continuous 
aggregate,  which  allows  to  pass  from  any  point  in  the  body  to 
any  other  through  any  third  point  belonging  to  the  body,  by  a 
continuous  path  lying  wholly  in  the  body  (^.  e.,  every  point  of 
which  belongs  to  the  body).  The  surface,  therefore,  is  also  con- 
tinuous, in  the  same  sense,  —  namely,  that  we  can  pass  by  con- 
tinuous motion  from  any  point  in  it  to  any  other,  through  any 
third,  by  a  path  every  point  of  which  is  in  the  surface.  More- 
over, the  moving  point  will  during  such  a  motion  remain  at  the 
given  distance  from  JL,  hence,  on  the  surface  of  an  imagined 
fixed  sphere ;  and  if  the  moving  point  is  conceived  to  be  fixed  in 
a  sphere,  of  same  center  A,  but  which  is  dragged  along  with  (B) 
in  its  motion,  we  see  that  every  portion  of  a  spherical  surface 
is  congruent  with  every  other  of  same  limits ;  that  is  to  say, 
the  spherical  surface  is  a  homogeneous  surface. 

Corollary  I. — Since  the  spherical  surface  separates  a  continu- 
ous portion  of  space  from  all  other  space,  that  is,  all  points 
that  can  be  reached  by  one  continuous  path  wholly  contained 
in  the  body,  from  all  points  of  space  that  can  not  be  reached 
by  a  continuous  motion  from  the  center,  unless  the  boundary 
of  the  body  is  crossed  by  a  path  of  which  a  portion,  at  least, 
does  not  belong  to  the  body,  —  it  follows  that  this  surface  is 
also  a  closed  surface. 

Definition  II.  —  All  points  in  the  sphere  (body  limited  by  the 
spherical  surface)  generated  by  the  given  radius  AB^  from  the 
given  center  A,  which  can  be  reached  by  continuous  motion  from 
the  center,  without  crossing  the  surface  at  all,  or  after  crossing  and 
recrossing  it  an  even  number  of  times,  are  said  to  be  within  the 
surface  or  inside  the  sphere,  and  the  distances  of  all  these  points 
from  the  center  (excluding  those  of  the  surface  itself)  are  said 
to  be  smaller  than  the  distance  AB.  All  points  that  can  be 
reached  by  continuous  motion  from  the  center,  only  after  crossing 
the  surface  once,  or  crossing  and  recrossing  it  an  odd  number  of 
times,  are  said  to  be  without  the  surface  or  outside  the  sphere,  and 
their  distances  from  A  are  said  to  be  greater  than  the  distance  AB, 

Corollary  II.  —  When  a  sphere  is  conceived  to  move  within 
the  boundaries  of  a  fixed  spherical  surface  which  is  always  in 
coincidence  with  the  surface  of  the  moving  sphere,  its  center 


(or  centers,  if  there  can  be  more  than  one),  remaining  at  a  con- 
stant distance  (or  constant  distances)  from  each  and  every  one 
of  the  same  fixed  aggregate  of  points  belonging  to  the  same 
surface,  will  remain  fixed  in  position.  But  since,  when  the 
sphere  moves  around  one  fixed  center,  from  which  it  is  con- 
ceived to  be  generated,  every  other  point  in  it,  at  a  distance 
from  the  center,  moves  upon  the  surface  of  a  sphere  (Theorem 
1),  it  follows  that,  given  a  spherical  surface  as  a  whole,  its  center 
is  uniquely  determined.  A  complete  spherical  surface  is,  there- 
fore,  said  to  he  the  locus  of  all  points  equidistant  from  a  unique 
fixed  pointy  called  the  center. 

Corollary  III. —  If  two  spheres  coincide  in  any  finite  portion 
of  their  surfaces,  they  coincide  throughout,  and,  hence,  have  the 
same  center  and  radius. 

For,  by  holding  the  finite  common  portion  of  each  spherical 
surface  fixed,  each  of  the  surfaces  remains  fixed  (Lemma  1). 
But  if  either  of  the  spheres  is  conceived  as  moving  within  its 
fixed  boundaries,  around  its  fixed  centre,  every  point  at  a  dis- 
tance from  the  fixed  centre,  moves  upon  a  corresponding  sphere, 
and  passes  into  the  position  of  every  possible  point  on  the  last, 
and  of  no  other  point.  Hence,  there  is  only  one  center  com- 
mon to  both,  since  the  portion  of  each  movable  spherical  sur- 
face, which  is  initially  in  coincidence  with  the  corresponding 
common  portion  of  the  two  fixed  surfaces,  can,  without  leaving 
either  of  the  spherical  surfaces,  pass  over  into  any  congruent 
portion  of  either. 

It  follows  from  what  has  preceded,  that  a  distance  is  a  geo- 
metrical magnitude.  No  two  points  at  different  distances  from 
a  given  point,  can  be  connected  with  the  given  point  by  the 
same  line,  which  might  serve  as  a  path  of  motion  from  one  end- 
point  of  each  distance  to  the  other.  If  we  take,  to  fix  ideas, 
as  the  connection  between  A  and  B,  a  line  of  some  determinate 
shape,  lying  wholly  within  the  sphere  described  by  (AB),  sl 
concentric  sphere,  described  by  a  distance  AC  less  than  AB, 
will  cut  every  line  {AB)  into  two  parts,  one  of  which  will  lie 
within  the  smaller  sphere,  and  the  other,  outside.  Thus  we 
see  that  to  a  smaller  distance  corresponds  a  smaller  path  of 
motion  from  one  end  of  the  distance  to  the  other.  This  sug- 
gests the  idea  of  representing  the  distance-magnitude  by  a  line. 
But,  then,  we  must  find  a  line  such  that  to  a  fixed  distance, 
from  one  fixed  point  on  it  to  another,  should  correspond  a 


67 

unique  position  of  the  line,  and  to  a  smaller  portion  of  the  line 
should  also  correspond  a  smaller  distance,  —  which  is  not  al- 
ways the  case  with  any  line.  Is  there  such  a  path  between 
A  and  jB?  I  say,  that  if  we  eliminate  all  paths  between  A 
and  B  which  can  assume  different  positions  while  A  and  B 
remain  fixed,  —  since  their  distance  is  unaltered,  —  there  must 
still  be  left  a  path  between  them  which  is  unique  for  this  fixed 
position  of  the  ends,  and  which  will  also  satisfy  the  other  re- 
quirement,—  as  I  shall  proceed  to  prove  with  the  utmost  rigor 
in  the  following  two  propositions  and  their  corollaries. 

Measuremefnt  of  Distances  From  a  Fixed  Origin  ;  Addition  and 
Subtraction  of  Given  Distances, 

Theorem  2.  —  Let  S  represent  a  sphere  described  by  radius 
a  from  origin  0.  OA  =  a,  as  soon  as  A  is  on  its  surface.  De- 
scribe from  A  a  sphere  S'  with  a  radius  6,  where  b  <^a.  Then 
0  is  outside  8'  (by  Defin.  2),  and,  hence,  some  portions  of  S  are 
outside  S';  for  if  the  whole  of  8  were  inside  8'j  the  center  of  8, 
which  is  separated  from  its  own  outside  by  some  portions  of  8y 
would  be  inside  8'  a  fortiori.  Similarly,  some  portions  of  8' 
are  outside  8;  otherwise  A  would  have  to  be  within  the  surface 
of  8f  and  not  on  it.  But  some  portions  of  8  are  also  inside  8';  for 
otherwise  it  would  be  impossible  to  reach  by  continuous  motion 
from  A  any  point  belonging  to  8,  before  crossing  the  surface  of 
8^  once,  —  which  is  absurd,  since  A  itself  is  on  the  surface  of  8, 
and  hence  belongs  to  >S';  a  portion  of  8  is,  therefore,  inside  8\ 
Also  some  portions  of  8'  are  inside  8 ;  for  if  8'  were  wholly 
outside  8y  it  could  not  contain  in  its  interior  any  point  belong- 
ing to  8 —  contrary  to  what  has  just  been  proved  to  be  the  case. 
Hence,  the  spheres  penetrate  each  other,  —  having  one  portion 
of  space  in  common,  limited  by  the  portions  of  their  surfaces 
which  they  have  inside  each  other,  and  two  other  portions  of 
space,  enclosed  each  between  the  interior  side  of  the  one  and 
the  exterior  of  the  other.  The  two  surfaces  must  intersect 
along  a  line  every  point  of  which  is  at  equal  distances  from  0 
and  from  A,  respectively.  Hence,  a  rigid  connection  of  any 
point  C  on  this  line,  with  both  centers  0  and  A,  will  be  cap- 
able of  displacing  itself  in  such  a  manner  that,  while  0  and  A 
remain  fixed,  C  shall  take  all  different  positions  on  the  line. 
Describe  now  from  0  a  sphere  8"  concentric  with  8y  with  a 
radius  equal  to  the  distance  from  0  of  a  point  lying  in  the  in- 


terior  of  both  8  and  S'.  Its  surface  will  pass  through  this 
point,  hence  will  cut  this  space  ^into  two  portions,  of  which 
only  one  will  lie  inside  S".  This  new  sphere  will  still  have  a 
portion  of  its  surface  inside  8'  and  the  other  portion  outside, 
since  it  encloses  centre  0,  which  lies  outside  8'.  Describing 
again  a  concentric  sphere  from  0,  with  a  radius  whose  end-point 
passes  now  through  a  point  lying  in  the  interior  of  8'  and  /S"', 
we  shall  still  more  reduce  the  portion  of  volume  enclosed  in  the 
interior  of  /S"  and  /S"',  to  that  only  which  is  left  inside  this  new 
sphere  8'"  and  8\  (The  portion  of  the  surface  of  8'  exposed  is 
continually  increasing  during  this  process  of  variation  of  >S',  8'\ 
/S""  •  •  •  etc.,  since  some  portions  of  it  enclosed  by  a  greater  sphere 
described  from  0  are  exterior  to  any  of  the  smaller  spheres.)  Evi- 
dently, by  continuing  this  process  far  enough  and  in  a  suitable 
manner,  the  variable  portion  of  volume  can  be  made  less  than 
any  assignable  small  volume.  This  will  happen  when  the  two 
spheres,  the  constant  one  8',  described  from  A,  and  the  variable 
8^'"\  described  from  0,  touch  each  other  in  one  or  several  points, 
or  in  one  or  several  lines ;  for  they  cannot  touch  in  any  portion 
of  surface  without  their  coinciding  in  every  part  (Cor.  3  to 
Theorem  1).  Now,  they  cannot  touch  in  several  distinct  lines 
or  distinct  points;  for  any  line  of  fixed  form,  connecting  0  and 
A  with  a  point  of  the  contact,  could  be  displaced  so,  that,  while 
0  and  A  remain  fixed,  the  point  on  the  line  which  has  been 
in  coincidence  with  the  touching  point  originally  taken,  shall 
coincide  with  any  other  point  in  the  common  touching  parts, 
which  must  be  at  the  same  distances  from  0  and  from  Ay  re- 
spectively. Hence,  since  by  the  principle  of  continuous  congru- 
ence in  motion  (Axiom  1),  the  figure  described  by  the  displaced 
line  is  a  continuous  one,  —  the  figure  described  by  a  determinate 
point  in  it  is  also  a  continuous  one.  It  must,  therefore,  be 
either  one  line,  —  any  portion  of  which  can,  by  a  continuous 
motion  along  itself,  pass  without  deformation  into  any  other  of 
same  limits,  —  or  one  point. 

Scholium. — By  continually  decreasing  the  radius  of  the 
variable  sphere,  and  dropping  the  condition  of  its  having  to 
pass  through  a  point  within  8 ',  we  shall,  at  last,  arrive  to  a 
series  of  spheres  which  have  all  their  volume  outside  8',  and 
have  not  one  point  in  common  with  the  constant  sphere.  It  is  evi- 
dent, therefore,  that  in  this  process  of  passing  continuously  from 
spheres  having  some  portions  inside  8',  to  such  whose  volume 


69 


rt^'o;^  o-^^^. 


and  surface  are  wholly  exterior  to  /S",  —  so  that  there  is  an  ap- 
preciable distance  from  each  and  every  point  belonging  to  the 
region  of  these  exterior  spheres,  and  each  and  every  point  be- 
longing to  the  region  of  S'j  —  we  must  on  the  way  encounter 
a  sphere,  such  that  any  sphere  described  by  a  radius  greater 
than  its  own,  has  some  portions  of  volume  inside  S',  and  any 
sphere  described  by  a  smaller  radius,  has  every  point  of  its  re- 
gion at  some  appreciable  distance  from  the  region  of  S'.  And 
in  crossing  from  the  series  of  greater  to  the  series  of  smaller 
spheres,  we  obviously  can  choose  two,  one  from  either  series, 
whose  radii  differ  as  little  as  we  please  from  each  other  and 
from  the  radius  of  the  bounding  sphere.  Evidently  the 
minimum  sphere  of  the  first  series  and  the  maximum  of  the 
second  series,  both  tend  towards  the  same  limit  —  the  bound- 
ing sphere,  which,  we  say,  touches  the  constant  sphere.  This 
bounding  sphere  must  have  at  least  one  point  in  common  with 
the  constant  sphere,  and  may  have  even  a  continuous  line  of 
contact,  until  otherwise  shown,  —  which  is  done  in  what  follows. 
Similar  remarks  hold  in  the  next 
supposition  of  the  present  demon- 
stration, viz.j  w^hen  we  have  to 
add  a  distance  to  a  given  one,  in- 
stead of  subtracting  (see  p.  70). 
Now,  the  contact  cannot  be  a 
line.  For  this  would  imply  that 
the  surface  of  the  sphere  >S'^"^, 
lying  wholly  outside  S'  according 
to  hypothesis,  is  divided  by  this 
line  into  two  parts  :  one,*  which 
together  with  the  exposed  por- 
tion of  >S"  that  has  been  continu- 
ally increasing  during  the  process 
of  variation  of  S^''\  makes  up  a 
closed  surface,  separating  the  in- 
terior of  both  spheres  from  the 
whole  of  the  exterior  space  ;  and 
the  second  part,  which,  as  derived 
from  the  portion  of  the  variable 
surface  that  has  remained  inside  the  combined  closed  surface  of 
both  spheres  during  all  the  process  of  variation,  must  still  be- 


FlG.  1,  a 


Fig.  la  and  lb  on  p.  70  illustrate  this  in  two  ways. 


70 


loDg  to  the  interior  and,  hence,  must  be  enclosed  between  the 
outside  of  8 '  and  the  interior  side  of  the  part  of  surface  of  8^'^'^ 
previously  considered.  Some  volume  must,  therefore,  be  enclosed 
between  the  outside  of  S'  and  the  outside  of  the  portion  of  sur- 
face of  /S^*^"^,  last  considered.  Describing  now  a  sphere  from  0 
with  a  radius  of  some  point  in  this  portion  of  volume,  which, 
consequently,  is  greater  than  the  radius  of  the  touching  sphere 
/S'^"^,  we  see  that  the  new  spherical  surface  must  enclose  every  point 
on  8^"^^  and,  hence,  also  the  line  of  contact.  It  must,  therefore, 
enter  into  the  interior  of  8'  along  a  line  lying  on  the  covered 
portion  of  8' ,  and  again  come  out  from  there  by  another  line, 
lying  on  the  exposed  portion  of  same,  —  which  is  possible. 
Hence,  the  contact  must  be  in  a  single  point. 

^rtjon  o/ j'^^ .  We  say  that  the  radius  of  /S'*^"\ 

described  from  0  and  touching 
8' ,  is  =  (a  —  6),  and  the  distance 
from  0  to  any  point  in  /S'*^'*^  and, 
in  particular,  to  the  point  of 
contact,  is  (a  —  h);  the  distance 
of  this  last  point  from  A  being 
equal  to  6,  the  distance  from  0 
to  A={a  —  h)-\-h  =  a  —  in 
perfect  agreement  with  original 
hypothesis. 

If,  instead  of  diminishing  the 
radius  of  the  sphere  /S,  we  should 
continuously  increase  it,  we 
would,  by  a  reasoning  perfectly 
similar  to  the  preceding,  come 
to  the  conclusion  that  the  space 
between  the  exterior  portion  of 
the  surface  of  8^""^  and  the  inte- 
rior of  8' y  would  continuously 
diminish  (and  likewise  the  exposed  portion  of  the  surface  of  8') 
and  could  be  made  less  than  any  assignable  volume,  —  when 
the  two  surfaces  would  have  to  touch,  either  along  a  single 
line,  or  in  a  single  point.  Now,  it  cannot  be  along  a  line ; 
for,  then,  —  since  /S^**^  in  the  limit  will  have  to  enclose  in  its 
interior  both  the  portion  of  8'  which  has  been  interior  to  8  at 
the  outset  and  which  has  kept  on  increasing  during  the  pro- 
cess of  variation  of  /S*^"^,  and  also  the  remaining  portion  of 


Fig.  1,  6 


71 

aS"  which  has  been  exterior  to  >S^"^  and  has  been  decreasing 
during  the  process,  —  there  must  be,  in  the  limit,  a  portion 
of  volume,  contained  between  the  portion  of  the  surface  of  S' 
considered  last,  on  the  outside  of  it,  and  that  portion  of  the 
surface  of  /S'^"^  which  is  now  separated  from  the  first  por- 
tion of  S'  by  the  surface  of  the  other  portion  of  S' .  The  two 
portions  of  surface,  enclosing  this  volume  and  belonging  to 
the  two  spheres  respectively,  are  separated  by  the  line  of  con- 
tact. Now,  if  we  describe  a  sphere  from  0  by  the  distance 
from  it  to  any  point  within  this  space,  this  sphere  will  lie 
wholly  within  8^""^  and  must  be  one  of  the  spheres  which  have 
cut  S'  before  reaching  the  limit.  It  must,  therefore,  first  emerge 
from  that  space  ;  but  as  it  cannot  cut  8^""^  it  must  cut  ;S"  in  two 
lines,  one  on  each  of  the  portions  of  its  surface,  separated  from 
each  other  by  the  line  of  contact  of /S"  and  8^''\  — which  is  im- 
possible. Hence,  the  contact  8'  and  8^""^  must  be  in  a  single 
point. 

We  then  say  that  the  radius  of  the  sphere  /S^"^  described  from 
0,  enclosing  8'  and  touching  it  in  one  point,  is  equal  to  a  +  6. 
The  distance  from  0  to  any  point  in  >S'^"^,  and,  in  particular,  to 
the  point  of  contact,  is  =  (a  -f  6) ;  the  distance  of  this  last  point 
from  A  being  equal  to  b,  the  distance  from  0  to  ^  is  (a  -j-  ^)  — 
6  =  a,  in  perfect  agreement  with  the  original  hypothesis. 

Suppose  now  we  have  to  add  to  OA  a  distance  AB  greater 
than  OA.  We  describe  then  from  A  a  sphere  8',  with  radius  A  B ; 
it  will  enclose  0  and,  hence,  at  least  a  portion  8 ;  it  may,  how- 
ever, enclose  it  all.  Describe  now  a  sphere  8^  from  0,  with  radius 
equal  to  AB  ;  it  will  enclose  A,  and  hence  8^  and  8'  must  have 
common  portions ;  but  neither  can  enclose  the  whole  of  the 
other,  since,  as  one  cannot  be  greater  than  the  other,  they  would 
have  to  coincide,  which  is  impossible — the  centres  being  distinct. 
Hence,  they  will  intersect  as  previously.  Increasing  now  the 
radius  of  the  sphere  described  from  0,  we  can  prove,  as  in  the 
last  case,  that  some  >S'*^''^  will  come  to  touch  8'  in  one  point  B, 
and  enclose  it  all,  so  that  OB  will  be  equal  to  OA  +  ABy  as 
previously.  Also,  by  diminishing  the  radius  of  8^  continuously, 
we  shall  get  another  point  of  contact,  whose  distance  from  0 
we  shall  call  OA  —  AB  =  a  —  b  <,0.  Such  a  point  of  con- 
tact as  this  last  can  never  be  obtained  from  the  above  rules  of 
addition  and  subtraction  by  adding  any  positive  distance,  either 


72 

greater  or  less  than  OAj  nor  by  subtracting  from  OA  any  posi- 
tive distance  less  than  OA.  Hence,  points  corresponding  to 
negative  distances  from  0,  where  OA  is  positive,  are  distinct 
not  only  from  one  another,  but  also  from  all  points  mark- 
ing positive  distances  from  0  under  same  hypothesis.  We  see 
now,  that  we  can  add  and  subtract  distances  as  abstract 
quantities. 

Scholium.  —  The  rules  for  addition  and  subtraction  given 
above  are  based  upon  the  fact  that  every  two  rigid  pairs  of  points 
connected  at  one  end  give  rise  to  an  infinity  of  distances 
between  the  free  ends,  having  a  perfectly  determinate  higher 
and  lower  limit  (maximum  and  minimum),  which  we  have  de- 
fined, respectively,  as  the  sum  and  the  difference  of  the  two 
given  distances  represented  by  the  rigid  pairs  themselves.  An 
additional  reason  for  singling  out  the  maximum  and  minimum 
distances  from  the  host  of  all  the  aggregate,  is  found  in  the 
further  fact  that,  when  the  ends  of  one  of  the  two  connected 
rigid  pairs  are  fixed  in  position,  the  position  of  the  remaining 
free  end  is  not  fixed  for  any  one  of  the  derived  distances,  with  the 
exception  of  these  two  limiting  distances.  It  is  also  evident, 
that  one  of  the  essential  conditions  which  the  addition  and  sub- 
traction of  measurable  quantities  must  satisfy,  namely,  that  the 
sum  increase  with  the  increase  of  each  of  the  terms,  and  that  any 
two  quantities  differing  in  value  should  always  give  a  determinate 
difference  (see  Weber,  "Traits  D'Alg^bre  Sup^rieure,"  1. 1,  p.  9), 
is  perfectly  satisfied  by  our  rules  for  all  the  three  cases  considered 
in  the  theorem.  In  order,  however,  that  these  rules  be  perfectly 
consistent,  and  should  lead  to  no  contradictions  in  their  appli- 
cation, it  is  further  necessary  to  prove  that,  in  the  first  place, 
the  operation  of  addition  obeys  the  associative  and  commutative 
laws  of  addition  of  abstract  quantities,  and,  in  the  second  place, 
the  rule  for  subtraction  is  actually  the  inverse  of  the  rule  for 
addition.  This  can  be  done  by  the  aid  of  the  following  two 
lemmas  : 

Lemma  1,  Theorem  3.  —  If  the  point  B  has  been  so  de- 
termined by  our  rule  of  addition,  from  the  fixed  points  0  and 
A,  that  OB  =  OA  -f  AB,  then,  taking  B  as  origin  and  a  mov- 
able rigid  pair  (AO')  congruent  with  (AO),  the  fixed  point 
Oj  determined  so  that  B0^=  BA  -\-  AO'  coincides  with  the 
original  starting  point  0. 


73 


Demonstration.  — Let  0,  Ay  B  be  the  triplet  of  points  of  the 
original  operation ;  S'^,  S'[,  aS"J,  the  corresponding  spheres. 
Then,  by  construction,  if  C  is  any  point  on  S'[  not  coincident 
with  Bj  we  have  OB  '>  OC.  The  spheres  S^  and  8'^  have  same 
centre  0  and  pass  through  A  and  J5,  respectively,  and  the 
sphere  8 'I,  with  center  Aj  is  wholly  interior  to  8^,  with  the 
exception  of  point  J5,  at  which  the  two  spheres  touch.  Let 
now  the  sphere  (-S'y),  conceived  rigid,  rotate  about  the  fixed 
point  A ;  every  point  (P)  belonging  to  the  rotating  sphere, 
will  remain  upon  a  sphere  described  about  J.  as  a  center, 
with  radius  (AP).  Hence,  the  center  of  the  moving  sphere, 
(0),  will  move  upon  the  surface  of  8^  described  from  A  with 
radius  (A  0)  ;  and,  since  (B)  is  the  only  point  of  the  surface  of 
(/S'q)  which  has,  at  the  beginning  of  the  motion,  been  at  a  dis- 
tance (AB)  from  A,  this  point  (^B)  will  be  the  only  point  in  the 
moving  surface  (8'^) 
which  will,  during  all  the 
rotation,  remain  upon  the 
surface  of  8'^.  All  other 
points  in  the  moving  sur- 
face (^o),  having  been  at 
the  beginning  of  the  rota- 
tion outside  iS"/,  will  re- 
main outside  during  any 
moment  of  the  rotation. 
Hence,  the  moving  sphere 
(81)  will,  during  all  the 
rotation,  remain  in  inte- 
rior contact  with  the  sta- 
tionary sphere  8'^,  touch- 
ing it  at  any  moment  in 
some  point  jB',  which 
marks  the  corresponding 
position  in  space,  at  that 
moment,  of  the  moving  point  (B)  rigidly  fixed  in  the  moving  sur- 
face of  {8q).  Let  0'  be  the  corresponding  position,  at  the  moment 
considered,  of  the  centre  (0).  It  is  now  evident  that  the  rotation 
of  (81)  can  be  so  arranged  that  its  centre  (0)  describe  the  whole 
of  the  spherical  surface  8 1.  (It  is  sufficient  for  this  purpose,  that 
the  radius  (AO)  describing  aS'J,  be  rigidly  fixed  in  the  sphere 
(8^),  which  moves  along  with  the  radius  about  A  fixed.)     Simi- 


FlG.  2. 


74 


larly,  (S'^)  can  rotate  so,  that  (B)  describe  the  whole  of  the 
surface  S'^.  Our  original  construction  of  the  triplet  of  points 
0,  A  J  and  Bj  will,  in  either  case,  at  any  moment  be  presented 
by  a  congruent  triplet  0',  Aj  B';  hence,  O'B'  =  O'A  +  AB'* 
We  conclude,  therefore,  that :  — 

First.  To  any  point  0'  on  the  surface  of  S^  corresponds  one, 
and  only  one,  point  B'  on  the  concentric  surface  S'^,  such  that 
0^S^=  WA  +  35"=  OB,  and,  at  the  sam^  time,  WJ  =  '0A 
and  AB'  =  AB,  and,  further,  0' B'  is  the  maximum  distance 
between  the  ends  of  the  connected  rigid  pairs  {0' A)  and 
{AB'). 

Second.  To  any  point  B'  on  the  surface  8 'I  corresponds  some 
point  0'  on  the  surface  /S'J,  such  that  our  original  construction, 
repeated  from  0'  and  A^  instead  of  0  and  Aj  will  lead  us  ex- 
actly to  the  point  B' .  We  cannot  say,  however,  until  proven 
so,  that  to  B'  corresponds  only  one  point  0' .  For,  although 
we  know  that  there  is  only  one  point  B'  on  aS"/,  corresponding 
to  a  fixed  point  0'  on  8\,  such  that  O'B'  >  0'  C —  C  being  any 
other  point  on  /S"/,  not  coincident  with  B',  we  cannot  affirm 

that,  conversely,  B' 0'  is 
also  the  maximum  dis- 
tance from  fixed  B'  to 
any  point  on  the  surface 
Sly  or,  in  other  words, 
that  BW>  B'D,  where 
D  is  any  point  on  S  \  not 
coincident  with  0'  ;  and 
if  it  is  not  the  maximum 
distance,  then  we  know 
from  the  preceding,  that 
a  spherical  surface  de- 
scribed from  B'  with  ra- 
dius =  B'O'y  will  cut  the 
surface  /S  J  in  a  closed  line, 
every  point  of  which  is 
exactly  at  the  same  dis- 
Ym.  2.  tances  from  B'  and  from 

J.  as  0',  and  can,  therefore,  replace  0'  in  our  construction. 
Let  us  now  start  with  the  fixed  points  B  and  Ay  as  in  the 
theorem,  and  let  the  movable  rigid  pair  {AO'),  congruent  with 


75 

(AO),  be  put  in  the  position  AO^y  where  BO^  —  BA  +  ^O^, 
in  the  sense  of  the  definition  of  addition  given  in  Theorem  2. 
I  say,  that  Oj,  which  is  by  construction  unique  on  S\,  and  which 
satisfies  the  inequality  B0^'>  BO' ,  where  0'  is  any  other  point 
on  S\  not  coincident  with  Oi,  cannot  be  any  other  point  than 
the  original  starting  point  0.  For,  let  it  be  some  other  point 
D 'j  D  \Q  unique  on  8\,  and  BD^  BO.  In  this  new  con- 
struction, B  being  the  starting  point  and  D,  the  final  point  in 
the  operation,  we  know  that,  when  (J5)  moves  upon  S  j,  its  cor- 
responding unique  point  (D)  moves  upon  S\,  (It  is  hardly 
necessary  to  explain  that  the  sphere  rotating  about  A  fixed  in 
the  new  construction,  and  corresponding  to  Sl'in  the  old  one, 
is  {S^l)  touching  internally  in  Z)  the  stationary  sphere  /SJ,  and 
supposed,  at  the  beginning  of  the  motion,  to  have  its  center  at  B.) 
We  know,  further,  that  to  every  point  (D)  on  S  \,  there  is  at 
least  one  point  {B)  (and  maybe  an  infinity  of  such  points)  on 
/Sj,  such  that  the  construction  repeated  from  the  point  (^),  or 
from  any  one  of  such  points,  and  from  A,  will  lead  exactly  to 
{D).  Let  now  (D)  move  up  to  0 ;  none  of  its  corresponding 
points  {B)  can,  for  this  new  position  of  (D),  coincide  with  B, 
since,  by  construction,  BD^  BO.  Hence,  the  point  corre- 
sponding to  0  in  the  new  construction,  must  be  some  other  point 
C  on  S'[,  —  and  we  have  CO  =  BD  >  BO  —  which  is  in  con- 
tradiction with  the  result  obtained  from  the  first  construction, 
namely,  OB  '>0C.  Hence  0^  marking  that  position  of  the  end  of 
the  rigid  pair  {AO')  in  the  new  construction,  which  corresponds 
to  the  maximum  distance  BO'  from  B  to  any  point  on  S\y 
cannot  coincide  with  any  point  ou  S\  except  0.  We  conclude 
hence,  that  if  OS  =  OA  -f  AB,  then  W)  =  ~BA -j-~AO  —  in 
the  sense  of  our  rule  of  addition,  —  and  if  the  distances  are 
measured  from  the  same  origin  0,  then  6  +  a  =  a  -f  6. 

Q.  E.  D. 


Lemma  2,  Theorem  4.  —  If  0^  =  OA  -\-  AB,  then  a 
sphere  described  from  B  with  radius  BA,  will  touch  externally 
the  sphere  described  from  0  with  radius  OA.  In  other  words, 
if  from  the  rule  of  addition  OA  —  OB  —  AB^  then  OA  is  the 
minimum  distance  between  the  extreme  ends,  of  the  fixed  rigid 
pair  {OB)  and  the  movable  rigid  pair  {BA)  jointed  to  the  first 
at  B ;  or,  otherwise  still,  the  rule  for  subtraction  is  an  actual 
inverse  of  the  rule  for  addition. 


76 


Demonstration.  —  Let  the  theorem  not  be  true.  Then  a 
construction  of  a  triplet  of  points,  0,  A,  and  B,  is  possible,  like 
that  in  the  figure,  where  the  sphere  S  3  —  touching  internally 
at  B  the  sphere  S'^  of  center  0,  and  having  its  center  A  on 
the  spherical  surface  S^j  likewise  described  from  0  —  has  a 
radius  AB  such,  that  the  sphere  8'^  described  from  B  with 
this  radius,  cuts  aS'^  in  a  closed  line  AA'.  Then,  we  can  describe 
from  B  as  center  another  sphere  S'^f  such  as  will  touch  exter- 
nally the  sphere  aS'^  in  the  point  C,  and,  consequently,  whose 
radius  BC<^BA,  Hence,  the  sphere  S^^  described  from  C 
with   radius    CD  =  ABy  having    B   and    some   space   in   the 

neighborhood,  in  its  inte- 
rior, cuts  S'q  or  encloses  it 
wholly  in  its  interior.  By 
conceiving  now  the  spher- 
ical shell  contained  between 
the  concentric  surfaces  S^ 
and  SI,  to  be  rigid  and  to 
rotate  about  the  stationary 
sphere  /S'^,  dragging  along 
(8^)  rigidly  fixed  to  this 
shell,  ((7)  will  come  to  A^ 
and  8^^  will  now  coincide 
with  a  congruent  sphere  8'[ 
described  from  A  with  ra- 
dius CD  —  AB  and  cut- 
ting S'q  in  a  closed  line  or 
enclosing  it  wholly  in  its 
interior ;  but  this  is  contrary  to  hypothesis,  according  to  which 
the  sphere  described  from  A  with  radius  AB,  is  8^,  tangent 
internally  to  8^.  Our  theorem,  therefore,  can  never  be  untrue, 
but  must  be  true  without  exception.  Hence,  the  rule  for  sub- 
traction is  a  real  consequence  of  the  rule  for  addition. 

Q.  E.  D. 
Scholium.  —  Now  we  can  prove  that  the  operation  of  addi- 
tion obeys  the  associative  and  commutative  laws  in  their  full 
generality.  We  have,  however,  to  remark  :  —  first,  that  from 
the  very  sense  of  the  rule  for  addition  follows  the  equality  : 
a  4-  6  +  c  =  (a  +  6)  -f  c,  and,  in  general,  any  number  of  terms 
following  the  first,  written  separately  with  signs  +,  may  be 


Fig.  3. 


OF  THE     * 

UNIVERSITY 


77 

incorporated  in  a  parenthesis  enclosing  the  first,  and  hence 
such  a  parenthesis  may  be  dropped ;  and  second,  that  in  any 
parenthesis,  by  Lemma  1,  we  may  interchange  the  order  of  the 
first  two  terms,  if  they  are  both  positive. 

Theorem  5.  —  The  operation  of  addition  obeys  the  associ- 
ative and  commutative  laws,  so  that  any  number  of  terms,  be- 
ginning from,  and  ending  with,  any  term  we  please,  may  be  en- 
closed in  a  parenthesis,  and  the  result  added  to  the  preceding 
terms,  —  and  any  term  may  be  transposed  forwards  and  back- 
wards, through  any  number  of  other  terms. 

Demonstration.  —  We  have  by  lemma  2  : 

6  +  c  —  c  —  6  =  &  —  6  =  0, 
and  also      6  +  c  —  (6  +  c)  =  (6  -f  c)  —  (6  -f  c)  =  0  ; 

.*.  —  (6  +  c)  =  —  c  —  6. 
Therefore,  a+6  +  c  —  (6-fc)  =  a  +  5  +  c  —  c  —  6 

...  a  -h  6  +  c  -  (6  +  c)  +  (6  +  c)  =  a  +  (6  +  c), 

or  a  +  6  -f  c  =  a  +  (6  +  c), 

and  still  otherwise, 

(a  +  6)  +  c  =  a  +  (6  +  c). 

Put  now  5  =  c?  -f  6  +  •  •  • , 

c  =  (/  +  ^)  +  •  •  •  +  (g  +  •  •  •  +  r)  4-  •  •  •  4-  < ; 
then  we  get 

a  +  (cZ  +  e  +...)  +  [(/+  ^)  +  ...  +  (5  +  ...  +  r)  +  ..-  +  q 

=  [a+((^+e+.'.  •)]  +  Kf+gH'  •  '+{{q+'  •  •+0+-  •  -+0] 


78 

=  («+^)  +  (e+-  •  •+/+^)  +  (-  •  •  +  ?+•  •  •+r+-  •  -  +  0 

=etc. 

Suppose,  now,  we  have  the  sum, 

and  we  wish  to  transpose  6  +  c  over  the  terms 
We  write  then, 

=  a+  [(6  +  c)  +  (cZ+6+/)] 

=  a  +  [(d^  +  e  +/)  +  (6  +  c)]  =  a  +  (d  +  e  +/+  6  +  c) 

We  add  then  the  remaining  terms  to  both  sides  of  the  equation, 
and  get  the  desired  transposition.  Q.  E.  D. 

Scholium.  —  The  process  of  obtaining  any  integral  multiple 
of  a  given  distance,  and  also,  of  obtaining  the  ratio  (commensur- 
able or  incommensurable)  of  two  given  distances,  ought  no 
longer  to  detain  us,  and  we  shall  only  remark  that  this  process 
is  a  direct  consequence  of  the  rules  for  addition  and  subtraction 
and  of  the  postulate  of  continuity,  which  says  that  between 
any  two  positions  of  any  geometrical  entity  (a  point  included) 
there  is  always  an  infinity  of  other  such  positions,  all  in  space, 
and  that  at  least  one  infinite  series  of  such  positions  must  be 
passed  through,  to  reach  either  of  the  two  given  positions  from 
the  other.  By  the  associative  law  of  addition  and  its  extension 
to  subtraction  in  considering  this  operation  as  the  addition  of  neg- 
ative terms,  we  can  get  a  given  series  of  points,  either  by  adding 
and  subtracting  separate  terms,  each  to,  or  from  the  preceding 
sum,  or  by  adding  to  the  same  given  term  a  series  of  distances 
equivalent  to  one,  two,  three  •  •  •,  n  •  •  •,  of  the  added  and  subtracted 
terms  taken  in  any  order  we  please.  This  last  process,  in  as- 
suming that  the  term  to  be  added  (in  a  positive  and  a  negative 
sense)  increases  continuously,  leads  us  to  the  important  concep- 
tion of  a  homogeneous  line,  completely  determined  by  two 


79 


points  in  it,  which  serves  as  the  basis  of  all  line-measurements, 
and  is  fully  described  in  the  following  definitions  and  succeed- 
ing corollaries. 

Definition  III. — By  conceiving  the  sphere  S'  described  from 
A  (in  Theorem  2),  to  vary  continuously  beginning  from  one 
whose  radius  is  indefinitely  small,  and,  passing  all  imaginable 
distances,  to  become  indefinitely  large,  we  shall  get  all  possible 
distances  OB,  positive  and  negative,  whose  general  expression 
is  {a  -f  x).  To  each  such  distance  from  0  corresponds,  by  con- 
struction, one,  and  only  one,  point,  and  the  aggregate  of  all  these 
points  will,  evidently,  form  a  continuous  line,  which,  satisfying 
fully  the  conditions  necessary  for  a  line  suitable  to  represent 
the  distance-magnitude,  may  be  called  the  distance-line  or  the 
straight  line. 


80 

Corollary  I.  —  To  every  distance  on  the  line,  as  measured 
from  0,  there  will  correspond  a  point  5  at  a  corresponding 
distance  from  A,  such  that  no  other  point  in  space  can  have 
the  same  distances  from  0  and  A,  respectively. 

For,  if  we  describe  two  spheres  from  0  and^^Jl  with  the  cor- 
responding distances  as  radii,  these  two  spheres  will  have 
only  one  point  in  common,  namely,  the  point  on  the  distance 
line  ;  but  if  there  were  another  point  in  space  at  the  same  dis- 
tances from  0  and  J.,  this  last  point  would  also  be  on  both 
these  spheres,  which  is  contrary  to  construction. 

Corollary  II.  —  If  J.  and  B^  and  A'  and  B' ,  are  two  pairs  of 
points,  at  equal  distances  from  each  other  singly,  then  the 
straight  lines  constructed  from  each  pair  as  from  0  and  A  or- 
iginally, will  coincide  with  each  other  along  the  whole  of  their 
extent  as  soon  as  the  congruent  pairs  {AB)  and  (^A'B')  are 
made  to  coincide.  This  follows  immediately  from  the  fact  that 
the  distance-line  is  unique  when  constructed  from  two  points 
of  fixed  position. 

Corollary  III. —  Every  point  not  on  the  distance-line  between 
A  and  B  or  its  prolongation,  is  such  that  we  can  find  a  con- 
tinuous series  of  points  of  which  this  is  one,  which  have  the 
same  distances  from  A  and  B,  respectively,  as  the  given  one. 

For,  since  the  given  point  is  not  on  the  distance-line  con- 
structed from  A  and  B,  it  follows  that  the  two  spheres  through 
the  point,  described  from  A  and  B  as  centers,  will  not  touch  in 
this  point,  but  cut  along  a  line  which  is  the  locus  of  points 
having  the  same  distances  from  A  and  B  as  the  given  one ; 
hence  our  corollary. 

Corollary  IV.  —  If  we  imagine  a  rigid  body  to  be  placed  so, 
that  two  of  its  points  coincide  with  A  and  B  considered  pre- 
viously, and  to  be  fixed  there,  then  all  the  points  of  the  body 
fall  in  either  of  the  following  two  categories  :  — 

(1)  A  continuous  series  of  them  coincide  with  corresponding 
points  on  the  distance-line  through  A  and  B,  which  is  fixed  in 
space  as  long  as  A  and  B  are  fixed  ; 

(2)  The  remaining  points  coincide,  each  with  a  point  in  space 
whose  distances  from  A  and  B^  respectively,  it  has  in  common 
with  every  point  on  the  intersection  of  the  two  spheres  described 
from  A  and  from  B  through  this  point. 

It  is  evident,  therefore,  that  any  point  in  the  body,  of  the 
second  category,  can  be  displaced  along  the  line  of  intersec- 


81 

tion  of  the  two  spheres,  on  which  it  originally  fell,  while  (A) 
and  (B)  are  still  fixed  in  A  and  By  letting  all  other  points  in 
the  body  take  care  of  themselves,  or,  rather,  the  rigidity  of  the 
body — which  consists  in  preserving  the  relative  distances  of  the 
points  of  the  body — take  care  of  their  individual  positions  dur- 
ing this  displacement.  We  find  then  that  every  other  point  of 
the  second  category  will  have,  in  all  its  displacements,  to  re- 
main on  a  corresponding  line  of  intersection  of  two  spheres 
from  A  and  B.  Every  point  in  the  first  category,  however, 
will  have  to  remain  stationary,  since  there  is  no  other  point  in 
space  having  the  same  distances  from  A  and  By  besides  the  one 
with  which  it  coincided  originally.  These  conditions  are  seen 
also  to  be  perfectly  compatible  with  the  relative  distances  of 
the  individual  points  of  the  body  from  one  another,  as  soon 
as  a  continuous  series  of  spheres,  described  from  A  and  B  as 
centers,  is  imagined,  together  with  their  mutual  intersections ; 
for,  as  the  points  of  the  body  move  along  these  lines  of  intersec- 
tion, these  lines,  together  with  their  spheres,  can  be  conceived 
to  glide  upon  themselves,  never  changing  their  form,  since  a 
sphere  is  a  homogeneous  surface ;  hence,  the  mutual  distances 
of  the  moving  points  are  defined  alike  throughout  their  motion. 
Such  a  motion  of  a  rigid  body,  about  a  stationary  straight  line 
(axis),  we  call  rotation^  and  the  body  moving  with  such  a  mo- 
tion, is  said  to  rotate  or  to  revolve  about  the  axis  ABj  or,  simply, 
about  the  two  fixed  points  A  and  B. 

Corollary  V.  —  Since  in  the  rotation  of  the  solid,  considered 
in  the  preceding  corollary,  any  two  points  in  the  solid,  (C)  and 
(i)),  which  lie  on  the  axis  {AB)y  will  remain  in  coincidence 
with  a  pair  of  congruent  points  C  and  I)  fixed  upon  the  dis- 
tance-line AB,  and  since  no  other  points  in  the  solid,  besides 
those  lying  on  the  axis,  remain  fixed, — it  follows  that  if  a  solid 
is  moved  so,  that  a  given  pair  of  points  in  it,  (C)  and  (D),  re- 
main fixed  in  space,  then  all  the  points  in  the  moving  solid  fall 
in  two  categories  :  —  such  as  remain  on  a  fixed  axis  determined 
by  C  and  D  and  constructed  from  any  two  'points  in  itj  A  and 
By  and  such  as  move  constantly  upon  corresponding  intersec- 
tions of  two  systems  of  spheres  of  centers  A  and  By  respectively. 

For,  rotating  upon  the  fixed  axis  constructed  from,  and 
therefore  determined  by,  C  and  D,  no  other  points  in  the  solid 
besides  those  lying  on  the  axis  can  remain  fixed  in  space.  Any 
two  points  [A)  and  {B)  in  the  axis,  however,  remaining  fixed,  the 


82 

solid  rotates  also  about  the  distance-line  constructed  from  A  and 
By  and  this  must  coincide  with  that  determined  by  C  and  D. 

Corollary  YI.  —  It  follows,*  that  there  is  no  difference  in  the 
form  of  the  distance-line  when  constructed  from  any  pair  of 
points  in  space,  and  that  any  two  such  lines  will  coincide  with 
each  other  throughout  the  whole  of  their  extent  as  soon  as  two 
congruent  points  in  both  are  made  to  coincide. 

Corollary  YII.  —  Any  portion  of  a  straight  line  will,  by  the 
preceding  corollary,  coincide  with  any  other  portion  of  the 
same  straight  line,  as  soon  as  their  limits  coincide.  In  other 
words,  a  straight  line  is  homogeneous,  i.  e.,  any  portion  can 
move  upon  the  whole  without  deformation. 

Corollary  VIII.  —  If  we  move  up  the  segment  (AB)  of  a 
straight  line  upon  itself  a  distance  AA'y  so  that  the  position  of 
(AB)  at  the  end  of  the  motion  will  be  A' BB' ,  then  the  whole 
line  AA' BB'  is  also  a  straight  line. 


A  a!  B  B' 

Fig.  5. 

In  fact,  AA' B  and  A' BB' ,  separately  considered,  being,  re- 
spectively, the  original  and  final  position  of  the  same  segment 
)AB),  are  two  segments  of  a  straight  line,  having  the  portion 
A' B  in  common.  Hence,  by  Corollary  YI,  they  must  coin- 
cide, each  with  a  corresponding  segment  upon  the  unique  dis- 
tance-line determined  by  A' B,  i.  e.,  AA' BB'  is  likewise  a  seg- 
ment of  a  straight  line. 

It  follows,  that  in  this  way  we  can  prolong  a  segment  of  a 
straight  line  indefinitely  far,  solely  by  shoving  it  along  itself 
and  its  successive  prolongations. 

Corollary  IX.  —  A  straight  line  cannot  have  more  than  one 
branch  on  each  side  of  a  point  belonging  to  it.  It  cannot,  for 
instance,  have  the  branch  ^^  on  one  side  of  the  point  B,  and 
J5(7and  BD  on  the  other  side  of  it. 


Fig.  6. 


*This  corollary  follows  also  very  readily  from  the  associative  law  of  ad- 
dition of  distances. 


83 

For,  otherwise,  by  revolving  a  solid  containing  both  branch 
ABD  and  branch  ABC,  about  ABD  fixed,  BC  would  be 
displaced  —  which  is  impossible  if  ABC  is  a  straight  line. 

Another  proof  is  obtained  thus  :  —  If  the  line  ABD  is  con- 
structed from  ABy  then  the  points  between  B  and  A  and  be- 
tween B  and  D  are  obtained  by  continuously  increasing  the 
distance  x  —  respectively  to  be  subtracted  from  and  added  to 
AB — from  zero  to  infinity.  But,  in  doing  so,  the  radius  of 
the  corresponding  sphere  8'  described  from  B  as  center,  on 
whose  surface  the  corresponding  points  lie,  increases  continu- 
ously, i.  e.j  these  points  recede  more  and  more  from 
the  center  By  both  ways,  and  can  never  come  back  to 
it  without  crossing  the  intermediate  spherical  surface, 
or,  which  is  the  same  thing,  without  retracing  their 
steps  backwards.  But  as  this  is  not  permissible  in 
the  continuous  description  of  the  distance-line,  we 
can  never  come  back  to  B,  Hence,  it  is  impossible 
that  the  additional  branch  BChe  ever  described. 

Theorem  6.  —  A  straight  line  ABy  issuing  from  a 
fixed  point  A  and  prolonged  indefinitely  on  the  other 
side  of  By  will  not  return  to  A  again,  after  any  num- 
ber of  prolongations,  each  equal  to  ABy  however  large 
that  number. 

For,  if  this  were  possible,  then  taking  just  half  of 
the  whole  extent,  whose  end  let  be  (7,  we  would  have 
two  distinct  straight  lines  between  A  and  (7,  namely, 
ABC  and  CB'Ay  since  CB'  A  is  supposed  to  be  differ- 
ent from  ABCy  as  the  point  B  is  not  supposed  to  re- 
trace its  steps  backwards  beginning  from  (7. 

Another  proof  is  exactly  like  the  second  proof  of 
Corollary  IX  to  Theorem  5,  where  the  impossibility 
of  returning  back  to  a  point  in  the  distance-line  is    Fia.  7. 
deduced  from  the  fact  that,  as  we  continually  move 
away  from  it  both  ways,  all  the  points  passed  in  either  sense, 
become  separated  from  those  not  yet  reached,  by  a  series  of 
closed  spherical  surfaces,  which,  by  the  definition  of  an  increas- 
ing distance,  can  never  be  crossed  backwards. 

Corollary. — From  this  theorem  and  from  the  preceding  one 
it  follows,  that  two  distinct  straight  lines  can  have  no  more  than 
one  point  in  common,  like  AOB  and  COD.  As  soon  as, 
besides   0,  another  point  E  in  CODy  is  made  to  coincide  with 


84 


E'  in  A  OB,  the  two  lines  must  coincide  up  to  the  ends  of 
the  smaller  segments  of  either,  on  both  sides  of  0,  and  their 
prolongations  must  coincide  as  far  as  we  please  to  take  them. 


Fig.  8. 

Definition  lY.  —  A  pair  of  straight  lines  having  only  one 
point  in  common  are  said  to  diverge  and  form  an  angle  at  the 
point  Oj  which  is  the  vertex  of  the  angle,  and  the  straight  lines 
themselves  are  the  sides  of  the  angle.  Such  a  pair  of  straight 
lines  is  called  a  crossing  pair  of  straight  lines. 

Scholium.  —  It  is  evident,  since  an  angle  has  four  segments, 
having  six  combinations  in  pairs,  if  we  leave  out  the  two  pairs 
belonging  to  the  same  straight  line  each,  we  get  only  four  —  the 
number  of  angles  as  defined  above.  (We  shall  learn  later  an 
extension  of  the  definition,  which  will  give  an  indefinite  num- 
ber of  angles.) 

We  shall  consider  at  present  one  angle,  made  only  by  two 
segments  of  distinct  straight  lines  issuing  only  on  one  side  of 
the  vertex  0,  like  A  OB.    We  imagine  this  to  be  a  rigid  figure, 

in  which  any  two  points 
k  and  I,  each  on  one  side, 
preserve  always  a  con- 
stant distance  between 
each  other. 

The    vertex   and    one 

side  of  any  other   angle 

A' 0' B'    can,    evidently, 

be  applied  to  the  vertex 

Fia.  9.  and  either  side  of  A  OB. 

If,  at  the  same  time,   a 

point  in  the  remaining  side  0' B'  can  he  made  to  coincide  with  a 

point  equidistant  from  0  on  OB — the  corresponding  free  side  of 

the  angle  AOB  —  without  breaking  the  rigidity  of  either  of 


86 


the  figures  as  explained  above,  the  whole  side  O'B'  will  coin- 
cide with  OB,  and  the  two  angles  will  he  said  to  he  equal;  if  on 
the  contrary y  this  is  not  possible,  the  angles  are  unequal. 

Let  us  see  whether  there  is  a  way  of  measuring  angles. 

Theorem  7.  —  If  two  straight  lines,  ^OCand  BOB,  inter- 
sect so  that  they  form  two  adjacent  angles  A  OB  and  BOG  (such 
as  have  one  side  in  common)  equal  to  each  other,  all  four 
angles  are  equal. 

Demonstration. — For,  put  an  exactly  equal  movable  figure, 
which  we  denote  by  small  letters  upon  the  given  one,  so  that 
ZaOh  of  the  movable  figure  shall  coincide  with  BOC  of  the 


c  b 


d     A 


c 
D 

Fig.  10. 


original  figure ;  then  Oc,  the  prolongation  of  aO,  will  coincide 
with  OD,  the  prolongation  of  BO,  and  Od,  the  prolongation 
of  60,  will  coincide  with  OA,  the  prolongation  of  CO. 

Hence  ZBOC=  ZaOh=  Z.hOG=  ICOD  =  /.cOd^^  ZDOA 
=  ZdOa=  Z  A  OB ;  that  is,  all  four  angles  are  equal. 

Q.  E.  D. 

Definition  V.  —  Each  of  the  four  equal  angles  formed  hy  the 
two  intersecting  lines,  is  called  a  right  angle,  and  the  lines  are  said 
to  he  perpendicular  to  each  other. 

Theorem  8.  —  Let  the  straight  lines  AOA'  and  BOB'  inter- 
sect each  other  at  right  angles.  Let  one  of  these  be  fixed,  and  the 
other  turn  round  0  in  space,  —  A  OA ' ,  say,  taking  up  all  possible 


86 


positions  compatible  with  the  rigidity  of  the  figure.  Then  OA 
will  generate  a  surface,  which  is  indefinite  in  extent  if  OA  is 
indefinite  in  extent,  and  which  is  capable  of  revolving  upon 
itself  around  0  without  deformation,  i.  e.,  any  portion  of  it 
inclosed  between  any  two  positions  of  OA^  like  OA^  and  OA^, 
will  be  capable  of  coincidence  with  any  other  portion  inclosed 
between  two  other  positions  of  OA  making  an  angle  equal  to 
AfiA^\  moreover,  if  BOB'  is  turned  over  about  the  fixed 
point  0,  so  that  the  segment  OB'  come  to  coincide  with  the 
original  position  of  OB,  and  OB  with  that  of  OB' ,  the  whole 
surface  will,  without  deformation,  turn  about  0  and  come  into 
coincidence  with  the  trace  of  its  original  position,  upside  down, 
as  soon  as  BOB'  comes  into  coincidence  with  B'  OB, 


*' 

B* 

Fig. 

11. 

./ 


Demonstration.  —  The  first  half  of  the  proposition  follows 
immediately  from  the  principle  of  continuity  of  congruence  in 
motion  (Axiom  1)  and  from  the  property  of  a  straight  line, 
combined  with  the  fact  that  in  the  rotation  of  a  rigid  body  any 
point  of  the  second  category  moves  upon  a  closed  line  —  the 
intersection  of  two  spheres  described  from  any  two  points  in 
the  fixed  axis  as  centers  and  passing  through  the  point  in 
question  (Corollary  TV.  to  Definition   3)*     The  second  half 

*  If  the  two  points  on  the  axis  are  taken  equidistant  from  0,  the  plane 
represents  the  system  of  concentric  circles  which  served  for  Lobatchevski  as 
a  definition  of  a  plane  in  his  work  (see  "  Urkundenznr  Geschichte  der  Nicht- 
Eukl.  Geom.,"  Engel,  1898,  pp.  93-109). 


87 


follows  from  the  equality  of  the  angles  A  OB  and  A  OB'.  For 
we  can  imagine  from  the  start,  that  a  duplicate  of  the  figure 
BOAOB'  has  been  brought  into  coincidence  with  B' OAOB  ; 
so  that  BOA  and  the  duplicate  of  B'  OA  generate  the  same  sur- 
face on  one  side,  as  the  duplicate  of  BOA  and  B' OA  itself y  on 
the  other  side.     Hence  the  theorem. 

Definition  YI.  —  The  surface  is  called  a  plane,  OA  —  the 
generator,  and,  in  any  of  its  fixed  positions,  a  half-ray,  which 
together  with  its  prolongation  on  the  other  side  of  0  composes 
a  complete  ray  or  element.  The  point  0  is  the  origin  ;  the  axis 
B'  OB  is  said  to  be  normal  to  the  plane  AOAy 

Corollary  I.  —  It  follows,  that  we  can  revolve  the  plane 
about  any  element  or  ray  passing  through  0,  until  its  upper 
side  comes  into  coincidence  with  the  original  lower  side,  and 


J^ 

^ 

Fig. 

11. 

y" 


/ 


^^4 


moe.  versa  —  the  ray  itself  remaining  fixed  ;  since,  in  this  mo- 
tion, the  ray  will  always  coincide  with  itself  while  the  nm-mal 
is  displaced  and  revolves  around  0  into  coincidence  with  its  own 
reversed  position. 

Corollary  II.  —  It  follows  also,  that  a  plane  divides  all  space 
into  two  equal  portions,  which  become  coincident  with  each 
other  as  soon  as  the  normal  coincides  with  its  reversed  position. 

Corollary  III.  —  The  plane  described  by  the  segment  OA, 
using  the  perpendicular  OB  as  an  axis,  is  identically  the  same 
as  that  described  by  the  prolongation  of  OA,  viz.,  OA'  ;  also 
this  plane  is  unique  as  long  as  BOB'  is  fixed. 


88 

Corollary  TV.  —  Given  the  plane  A  OA^  whose  normal  is 
OB,  and  a  straight  line  OC  not  lying  in  this  plane,  then 
Z.BOC  is  not  a  right  angle. 

For,  joining  by  a  straight  line,  h  and  a,  a  point  on  the  normal 
and  some  point  in  the  plane,  not  0,  and  then  revolving  {BhOci) 
around  Bh  0  fixed,  ihd)  must  somewhere  in  its  motion  intercept, 
in  some  point  c,  the  straight  line  OC,  which  is  supposed  to  re- 
main stationary  until  this  occurs ;  since,  after  ( Oa)  comes 
round  back  to  its  original  position,  (60a)  will  have  described 
a  closed  surface,  separating  a  finite  portion  of  space  from  all 
other  space.  This  surface  will  consist,  partly  of  the  portion  of 
the  plane  described  by  the  segment  {Oa)  of  the  element  OA^ 
in  which  a  is  situated,  and  partly  of  the  lateral  surface  de- 
scribed by  (ah),  every  point  of  which  will  be  at  some  appreci- 
able distance  from  0.  Hence,  a  sphere  described  by  the 
smallest  distance  from  0  of  any  point  in  the  lateral  surface, 


Fig.  12. 

will  enclose  an  appreciable  portion  of  OC;  while  a  sphere  de- 
scribed by  a  distance  greater  than  that  of  the  point  in  the  lateral 
surface  farthest  from  0  (only  the  finite  segment  (ah)  is  con- 
sidered), will  enclose  in  its  interior  a  portion  of  OC  having 
some  points  exterior  to  the  closed  surface.  Hence,  a  variable 
point  on  OC,  in  moving  continuously  from  a  position  on  the 
interior  portion  towards  a  position  on  the  exterior  portion, 
must  cross  the  closed  surface.  But  as  0  C  can  meet  neither  an 
element  of  the  plane  A  Oa,  nor  the  normal  OB,  in  any  other 


89 

point  than  0,  it  must  pierce  the  lateral  surface  described  hj 
(ba)  in  some  point  c.  Thus,  the  generator  of  the  lateral  sur- 
face (ba)  in  the  position  of  bca  intercepts  OC  in  the  point  c. 
Let  then  the  element  (OA^))  starting  from  its  new  position,  de- 
scribe its  own  plane  anew,  dragging  along  OC  in  its  motion. 
Then,  this  last  will  describe  a  surface  which  will  lie  wholly  on 
that  side  of  the  plane  A  OA^  where  the  half  of  the  normal  OB 
is  situated.  Reversing  now  the  plane,  the  surface  generated 
by  OC  will  lie  wholly  on  the  side  where  OB'  was  originally, 
and  will  be  separated  from  its  original  position  by  double  the 
space  between  its  original  position  and  the  plane  A  OAy  There- 
fore, the  surface  generated  by  0(7  in  its  rotation  around  the 
axis  BOB',  is  not  a  plane  (Corollary  II),  and  hence  Z.BOC  is 
not  a  right  angle. 

Corollary  V.  —  It  follows,  that  all  planes  coincide  with  one 
another  as  soon  as  their  normals  and  origins  coincide. 

Theorem  9.  —  All  right  angles  are  equal. 

Demonstration.  —  If  the  vertex  of  any  right  angle  is  made 
to  coincide  with  the  origin  0  of  our  plane,  and  one  of  its  sides, 
with  the  normal  OB  of  the  plane,  the  other  side  must  coincide 
with  some  one  of  the  elements  of  the  plane,  say  OA^,  by  pre- 
ceding corollary.  Hence,  all  right  angles  can  be  made  to  coin- 
cide, or  are  equal.  Q.  E.  D. 

The  preceding  theorems  about  the  properties  of  a  plane  and 
of  right  angles  will  suffice  to  render  more  concrete  our  notions 
about  angles  in  general  as  geometrical  magnitudes.  First,  we 
observe  that  we  have  found  a  natural  unit  for  measuring  angles, 
namely,  the  right  angle ;  and,  secondly,  we  shall  soon  see  how 
the  plane  will  afford  us  a  means  of  comparing  all  possible 
angles  with  our  standard  unit,  the  right  angle.  The  following 
preliminary  remarks  are  necessary. 

Scholium  I.  —  Any  pair  of  crossing  lines  will  be  capable  of 
being  applied  to  the  plane  so,  that  the  vertex  shall  lie  on  the 
origin,  and  the  lines  themselves  shall  lie  wholly  in  the  plane. 

In  fact,  one  of  the  two  lines  can  always  be  brought  into  co- 
incidence with  any  ray  in  the  plane  so,  that  the  vertex  fall  on 
the  origin.  If,  now,  the  other  line  falls  upon  another  ray,  the 
proposition  is  proved  for  the  case  under  consideration.  If  not, 
this  other  line  will  have  pierced  the  plane  (which  is  admissible 
during  the  process  of  applying ;  we  may  imagine,  if  necessary, 


90 

a  portion  of  the  plane  removed  during  the  application  and  then 
restored  back  to  its  original  position ;  see  also  definition  of 
rigidity),  and  will  have  only  one  point  in  common  with  the 
plane,  viz.,  the  origin,  like  OB  or  00  in  Fig.  12.  If,  now, 
we  revolve  the  plane  about  the  fixed  ray  with  which  the  first 
side  is  in  coincidence,  it  will  sweep  through  all  space  during  a 
revolution  that  brings  the  normal  into  its  reversed  position. 
Hence,  some  time  during  this  revolution,  it  will  have  to  inter- 
cept another  point  on  the  line  which  has  not  been  in  it  origi- 
nally and  which  is  supposed  to  have  remained  fixed  in  position  ; 
but  just  as  this  interception  occurs,  this  line  will  have  two 
points  in  common  with  some  ray  in  the  plane,  and  will,  hence, 
coincide  with  it  along  the  whole  of  its  extent  —  i.  e.,  both  lines 
of  the  crossing  pair  will  lie  in  the  plane.  Hence,  all  pairs  of 
crossing  lines  can  be  made  to  coincide,  each  with  a  pair  of 
crossing  rays  in  the  plane.  In  other  w^ords,  all  possible  angles 
are  to  be  found  among  the  different  angles  between  the  different 
rays  of  any  plane. 

Scholium  II.  —  If  we  apply  any  two  perpendicular  lines  of 
indefinite  extent  to  a  corresponding  pair  of  rays  in  the  plane, 
we  see  that  the  whole  extent  of  the  plane  can,  from  any  initial 
position  of  a  ray,  OA  say,  be  divided  into  four  congruent  parts, 
each  of  which  will  be  enclosed  by  a  right  angle.  The  end  of 
any  fixed  distance  from  0,  measured  along  a  ray,  will  generate 
in  its  motion  around  0  a  homogeneous  curve  every  point  of 
which  is  equidistant  from  0.  The  curve  is  called  the  circum- 
ference of  a  circle,  and  0  is  its  center.  A  portion  of  the  plane 
enclosed  between  any  two  rays  measured  one  way  from  center 
to  circumference  and  termed  radii  is  called  a  sector ^  that  is,  a 
part  of  the  whole  portion  of  the  plane  enclosed  by  the  circumfer- 
ence ;  such  a  sector  is  congruent  with  any  other  sector  enclosed 
between  two  radii  making  the  same  angle  with  each  other.  It 
is  now  evident  that,  since,  whenever  the  angle  between  two 
fixed  radii  is  equal  to  the  angle  between  two  other  fixed  radii, 
both  the  sectors  of  the  circle  and  the  segments  of  the  circumfer- 
encCy  or  arcs,  enclosed  by  the  respective  pairs  of  radii,  are  equal 
each  to  each  —  the  angles  can  be  measured  either  by  the  cor- 
responding sectors  or  by  the  arcs  enclosed  between  their  sides, 
so  that  a  greater  angle  corresponds  to  a  greater  sector  or  arc,  and 
a  multiple  or  part  of  an  angle  corresponds  to  the  same  multiple 
or  part  of  the  corresponding  sector  or  arc.     Since  every  possible 


91 


position  of  two  intersecting  straight  lines,  with  respect  to  each 
other,  has  been  proved  to  find  its  analogue  in  some  position  of 
two  intersecting  rays  of  a  plane,  it  is  sufficient  to  investigate 
these  last.  Now,  we  can  conceive  the  whole  aggregate  of  dis- 
placements possible  for  a  ray  in  the  plane  (Fig.  13)  with  re- 
spect to  a  fixed  ray  A  'a'  OaA,  to  be  bound  up  with  the  corre- 
sponding displacements  of  a  circle  rigidly  fixed  to  the  moving 
ray  and  revolving  with  it  about  the  center,  so  that  the  circum- 
ference moves  in  its  own  trace.  Then  we  see  that,  if  (Oc),  one 
segment  of  the  moving  ray,  makes  in  any  of  its  positions  an 
angle  AOq  with  OJ.,  measured  by  the  arc — passed  over  by  the 


moving  point  in  the  circumference  —  from  a  to  c,  then  the  pro- 
longation of  (Oc)  must  have  been  displaced  just  as  much  on  the 
other  side  of  OA' — the  prolongation  of  OA — and  must  form 
an  equal  angle  A'  OC — as  measured  from  the  fixed  segment 
OA'  or,  along  the  circumference  of  the  circle,  from  a'  to  c',  in 
the  same  sense  as  ac.  For,  any  fixed  point  in  the  moving  cir- 
cumference must  have  been  displaced  an  equal  arc  with  any 
other.  So  we  see  that  the  two  half-rays  of  the  same  ray  lie  on 
opposite  sides  of  the  two  half-rays  of  any  other  ray  not  coinci- 
dent with  it  —  since  any  fixed  ray  can  be  taken  as  the  initial  ray 
—  and  the  angle  made  by  the  prolongations  of  any  two  half-rays 
will  be  exactly  equal  to  the  angle  made  by  the  half-rays  them- 
selves.    Also,  that  any  point  in  the  moving  ray  will  be  dis- 


92 


placed  a  quarter,  a  half,  three  quarters,  and  a  whole  circumfer- 
ence, when  the  ray  is  displaced  one,  two,  three,  or  four,  right 
angles ;  further,  that  the  half-ray  will  fall  upon  its  own  prolon- 
gation —  for  a  displacement  of  two  right  angles,  and  will  come 
to  its  original  position — for  a  displacement  of  four  right  angles. 
It  follows,  then,  that  a  half-ray,  in  any  of  its  positions,  makes 
with  the  two  half-rays  of  another  ray  two  angles,  one  of  which 
is  just  as  much  less  than  a  right  angle,  as  the  other  is  greater,  — 


their  sum  being  equal  to  two  right  angles  {acute  and  ohtvLse 
angles). 

The  following  theorem,  concerning  angles  in  space,  becomes 
at  once  evident  by  the  application  of  the  angles  to  the  plane. 

Theorem  10.  —  Two  adjacent*  angles  whose  two  non-com- 
mon sides  form  the  opposite  prolongations  from  the  vertex  of 
the  same  straight  line,  are  together  equal  to  two  right  angles ; 
the  two  non-adjacent  angles,  made  by  two  straight  lines  and 
their  prolongations,  respectively,  are  equal ;  and  all  the  four 
angles  taken  together  are  equal  to  four  right  angles. 

Definition  VII.  —  A  figure  bounded  by  three  straight  lines 
intersecting,  two  by  two,  in  three  points  and  making  three 
angles,  each  less  than  two  right  angles,  is  called  a  triangle. 
We  must  not,  at  the  beginning,  consider  the  triangle  as  contain- 
ing any  surface  that  may  be  limited  by  the  sides  of  the  triangle. 

*  Adjacent  angles  are  defined  as  usual. 


93 

Scholium.  —  It  is  evident  from  the  way  we  have  constructed 
the  straight  line,  that  any  three  points  at  fixed  distances  from 
each  other  are  capable  of  congruence  with  any  other  triplet  of 
points  of  the  same  fixed  distances ;  since,  if  in  any  such  a  triad 
we  revolve  one  of  the  points  around  the  straight  line  connect- 
ing the  remaining  pair,  we  get  the  locus  of  all  the  points  which 
are  at  the  same  two  distances  from  the  fixed  pair  as  the  given 
third  one,  and  no  point  outside  this  locus  can  have  the  same 
distances  from  the  fixed  pair,  —  as  shown  in  Theorem  2-5  and 
corollaries.  Hence,  when  the  corresponding  points  of  the  two 
equidistant  pairs  in  the  two  triads  are  brought  into  coincidence, 
the  third  ones,  in  each  triad,  will  be  capable  of  coincidence. 

Corollary.  —  Since  as  soon  as  the  ends  of  two  equal  distances 
coincide  the  straight  lines  representing  these  distances  coincide, 
it  follows  that  any  two  triangles  whose  sides  are  respectively 
equal  to  each  other  are  congruent. 

Theorem  11.  —  Two  triangles  are  equal  when  two  sides  and 
the  included  angle  of  one  are  respectively  equal  to  two  sides 
and  the  included  angle  of  the  second. 

The  demonstration  given  by  Euclid  in  his  Elements  for 
this  theorem,  holds  here  word  for  word,  and  need  not  be 
repeated. 

Corollary.  —  In  every  isosceles  triangle  the  angles  at  the  base 
(opposite  the  equal  sides)  are  equal.  For,  its  duplicate  can  be 
applied  to  it  so,  that  the  equal  sides  be  interchanged  by  turning 
over ;  the  base  and,  hence,  the  angles  interchanged,  will  still 
coincide  with  the  corresponding  ones  in  the  original  triangle. 

These  two  theorems  concerning  congruent  angles  and  tri- 
angles are  sufficient  to  deduce  a  most  fundamental  property  of 
the  plane — namely,  that  the  origin  may  be  transposed  to  any 
point  in  the  plane,  and  hence,  any  straight  line  having  only 
two  points  in  common  with  the  plane,  anywhere,  will  lie  wholly 
in  the  plane ;  hence,  the  plane  itself  is  capable  of  translation  or 
rotation  upon  itself  without  deformation. 

Theorem  12.  —  A  straight  line  having  two  points  in  common 
with  a  plane,  will  lie  wholly  in  that  plane. 

Demonstration.  —  If  the  two  points  are  upon  the  same  ray, 
the  straight  line  coincides  with  the  ray,  i.  e.,  lies  in  the  plane. 
Suppose,  however,  that  the  two  points  lie  upon  different  rays ; 
we  then  can  prove  that  any  other  point  of  the  straight  line  lies 
upon  some  other  ray  of  the  plane. 


94 

Let  BOB' ,  where  OB  =  OB' ^  be  the  normal  of  the  plane 
AOC,  and  let  the  straight  line  X'X cut  the  rays  OA  and  OC 
in  the  points  A  and  C,  respectively ;  we  have  to  prove  that  any 
point  jD  of  the  line  X' X  lies  upon  a  ray  OD.  Connecting  OD, 
we  join  B  and  B' y  respectively,  with  Ay  C,  and  I), 


K~~~-~~~^. 


Fig.  14. 

Then  ABOC^AB'OC  and  ABOA  =  AB'OA,  by  last 
proposition. 

.'.BC=:B'C,BA  =  B'Ay  whence  AABC=  AAB'Cy  by 
corollary  to  Definition  VII. 

Whence,  Z.BAD=  AB'AJD; 

.  • .  ABAD  =  AB'ADy  and  ^D  =  J5'D ; 

whence,  again,  ABOD  =  AB' OD, 

.  • .  ABOD  =  right  angle  by  Theorem  10. 

OD  is,  therefore,  a  ray.     Now,  since  this  is  true  for  any  point 
D  in  the  line  X'X,  the  whole  of  its  extent  lies  in  the  plane. 

Q.  E.  D. 

Definition  YIII. — A  plane  can  now  be  re-defined  as  the  sur- 
face in  which  every  straight  line  lies  wholly  if  it  passes  through 
two  of  its  points. 

Corollary.  —  It  immediately  follows,  that  any  angle  can  be 
made  to  lie  anywhere  in  the  plane ;  and  so  also  a  triangle,  —  for 
one  side  of  the  triangle  will  lie  wholly  in  the  plane  as  soon  as 


95 

its  two  ends  lie  in  the  plane ;  now,  if  also  the  third  vertex  is 
brought  into  coincidence  with  some  point  in  the  plane,  by  re- 
volving the  triangle  around  the  side  held  fixed  in  the  plane, 
the  other  two  sides  will  likewise  come  wholly  into  the  plane. 

Scholium.  —  When  a  movable  half-ray  in  a  plane  describes 
a  positive  (continuously  increasing)  angle,  and  in  doing  so  it 
slides  upon  a  fixed  straight  line  termed  transversal  and  inter- 
secting its  initial  position  in  a  point  other  than  the  vertex,  the 
segments  which  it  cuts  off  upon  this  transversal  —  measured 
from  the  fixed  point  of  intersection  towards  the  corresponding 
positions  of  the  variable  point  of  intersection  with  the  moving 
half-ray  —  will  continuously  increase  as  long  as  the  moving 
half-ray  meets  the  transversal ;  that  is,  until  a  segment  is 
reached  which  exceeds  in  length  any  arbitrarily  given  finite  seg- 
ment, no  matter  how  great.  For,  any  point  of  the  transversal, 
that  is  at  a  finite  distance  greater  than  the  given  length  from 
the  origin  of  the  segments,  can  readily  be  joined  with  the  ver- 
tex of  the  variable  angle,  and  will  therefore  form  the  limit  of 
one  of  the  segments  greater  than  the  given  one. 

Definition  IX.  —  The  normal  is  said  to  be  perpendicular  to 
the  plane,  because  it  is  perpendicular  to  every  straight  line 
passing  through  it  and  lying  wholly  in  the  plane. 

Theorem  13.  —  A  straight  line  perpendicular  to  any  two 
intersecting  straight  lines  at  their  point  of  intersection,  is  per- 
pendicular to  the  plane  in  which  these  straight  lines  are  situ- 
ated. The  straight  lines  need  not  be  the  rays  of  the  original 
construction  of  the  plane,  but  any  two  straight  lines  intersect- 
ing   these    and,    hence, 

^  M 


\ 
\ 
\ 
\ 


lying  in  their  plane.  L — 

The  proof  is  word  for 
word  that  given  by  Eu- 
clid in  Book  XI,  prop- 
osition lY,  of  his  Ele- 
ments. 

Corollary  I. — An  im- 
mediate consequence  is 
that  at  every  point  of  a  ^' 

plane  we  can  erect  a  per-  Fia.  15. 

pendicular  to  the  plane. 

For,  if  K  is  such  a  point,  not  the  origin  of  the  original 
construction,  and  LKM  and  KN  are  any  two  perpendicular 


\' 


96 

straight  lines  through  K  in  the  same  plane,  then,  fixing  the 
line  LKM  and  revolving  {NK)  about  it  the  amount  of  one 
quadrant,  into  the  position  of  iV^' -ST,  this  last  is  now  perpen- 
dicular to  KM  and  KN)  hence,  it  is  perpendicular  to  the  plane 
in  which  they  are  situated.  In  other  words,  KN'  may  now  be 
regarded  as  a  normal,  and  all  straight  lines  in  the  given  plane 
through  Kj  as  the  rays. 

Corollary  II. — Since  any  two  planes  coincide  as  soon  as  their 
origins  and  normals  coincide,  it  follows  that  a  plane  will  coin- 
cide with  itself  when  the  origin  is  displaced,  in  any  manner 
whatever,  to  any  other  point  in  the  same  plane,  provided  the 
normal  at  the  origin  is  made  to  coincide  with  the  normal  at  the 
point  to  which  the  origin  is  shifted ;  also,  that  the  origin  or, 
indeed,  any  point  in  a  plane,  treated  as  such,  may  displace  itself 
continuously,  describing  any  curve  in  the  plane  —  the  whole 
plane  remaining  unaltered  in  shape  or  position ;  and,  further, 
that  the  plane  may  be  turned  upside  down,  shifted  upon  itself 
in  any  manner  whatever,  without  altering  its  position  or  shape 
as  a  whole  (Leibnitz). 

At  this  stage  of  our  investigation  our  elementary  figures,  viz., 
the  angle,  the  triangle,  and  the  circle,  can  be  made  more  concrete. 
Since  each  of  these  figures  can  be  made  to  lie  wholly  in  a  plane, 
we  may  suppose  their  boundaries  to  limit  corresponding  por- 
tions of  a  plane.  This  is  the  reason  of  their  being  called  plane 
figures,  in  contradistinction  to  those  which  cannot  be  made 
to  lie  wholly  in  a  plane.  Thus,  a  network  of  straight  lines 
and  circles  may  be  conceived  to  cover  these  plane  figures,  just 
like  the  plane  itself ;  crossing  and  recrossing  one  another  in  all 
possible  ways,  since  every  point  in  a  plane  may  be  conceived  as 
an  origin  of  rays  and  angles  and  as  a  center  for  the  circumfer- 
ences of  the  circles  described  by  the  fixed  points  in  these  rays. 
Geometry  invariably  makes  use  of  this  conception  of  the  plane, 
in  the  way  of  d^jiat :  "  Describe  a  circle,  from  such  and  such  a 
point  as  centre,  and  with  such  and  such  a  distance  as  radius,^' 
etc., — just  as  it  does  with  respect  to  homogeneous  space  in 
general. 


CHAPTER  III 

THE  QUADRILATERAL  AND  THE  IMMATERIAL  QUADRILATERAL. 
PARALLEL  STRAIGHT  LINES 


The  following  propositions  of  the  first  book  of  Euclid's  Ele- 
ments can  now  be  proved  very  rigorously,  either  by  Euclid's 
demonstrations,  or  in  a  more  elegant  way  —  like  the  one  used  by 
Legendre  and  his  followers,  —  since  none  of  the  hypotheses, 
whether  tacit  or  explicit,  which  Euclid  assumes  in  the  shape  of 
postulates,  definitions,  and  axioms,  in  these  demonstrations,  are 
now  wanting  a  solid  basis.  I  omit  the  demonstrations,  refer- 
ring the  reader  to  Euclid  or  Legendre,  whose  treatment  of  these 
propositions  is  admitted  to  be  rigorous  and  faultless,  once  you 
grant  the  postulates  and  axioms  upon  which  their  proofs  rest, 
and  which  have  now  been  established  with  the  utmost  rigor. 
The  propositions  referred  to  are  : 

IX*,  X,  XI,  XII,  XIV,  the  converse  of  XY,  XVI,  XVII, 
(for  the  last  two  I  prefer  to  give  my  own  proofs,  which  will 
throw  some  light  on  the  nature  of  parallels);  then  XVIII, 
XIX,t  XXI,  XXII,  XXIII,  XXIV,  XXV,  XXVI.  Now 
we  come  to  Euclid's  treatment  of  parallels,  which  has  been 
acknowledged  to  be  the  weakest  spot  in  the  Euclidian  geometry. 
I  propose  to  treat  this  subject  in  an  entirely  difierent  manner, 
using  again,  as  in  the  rest  of  this  treatise,  the  kinematic  method, 
which  is  much  more  powerful  than  the  static  method  adopted 
by  Euclid,  and  which,  I  flatter  myself  with  the  belief,  will  es- 
tablish on  a  foundation  firmer  than  ever  before,  the  most  im- 

*  The  propositions  left  out  have  been  proved  by  ns  explicitly  or  implicitly, 
excepting  VII,  which  is  unnecessary,  and  VI,  which  is  a  consequence  of  XXVI, 
case  1,  since  the  duplicate  of  a  triangle  with  two  equal  angles  can  be  turned 
over  and  applied  to  the  original  triangle,  with  which  it  will  coincide. 

fThe  Twentieth  of  Euclid  and  also  the  more  general  proposition  that  a 
straight  line  is  the  shortest  path  between  two  points  (not  the  shortest  dis- 
tance, since  there  is  only  one  distance  between  two  fixed  points)  can  be 
proved  immediately  by  the  consideration, —  that  any  point  of  the  second  cate- 
gory with  respect  to  two  fixed  points,  lies  upon  the  intersection  of  two 
spheres  described  from  the  points  with  radii  whose  sum  is  greater  than  the 
sum  of  the  radii  of  the  two  spheres  in  contact  described  from  the  same 
fixed  points,  whose  point  of  contact  is  a  point  of  the  first  category,  on  the 
straight  line  between  the  fixed  points,  by  the  very  construction. 

97 


98 


portant  truths  of  similarity  and  proportion,  on  which  rests  all 
the  grand  superstructure  of  our  actual  mathematics,  both  pure 
and  applied. 

I  proceed  to  prove  XVI  and  XYII  of  Euclid's  first  book. 

Theorem  14.  — The  sum  of  any  two  interior  angles  of 
a  triangle  is  less  than  two  right  angles. 

a  +  b<2rt  Z's, 

Demonstration.  —  If  a  +  6  ^  2  rt.  Z's,  then,  because  /.ABE 
-f  6  =  2  rt.  Z's  (Theorem  10),  it  follows  a  +  6  ^Z.ABE-\-  h, 
and  a  ^  /.ABE.  For  a  similar  reason  h  ^  /.DAB.  Now,  apply- 
ing the  lower  part  of  the  diagram, 
namely  DABEy  to  the  upper  ABC 
so,  that  A  shall  fall  upon  B  and  B 
upon  Ay  and  the  angle  a,  being 
^  /EBAy  shall  coincide  with  or  in- 
close the  latter,  and  b  shall  coincide 
with  or  inclose  /DAB,  —  AD  will 
lie,  either  on  the  side  BC,  or  within 
the  angle  6,  and  BE,  either  on  A  (7,  or 
within  the  angle  a.  In  either  case  AD 
and  BE  will  intersect  —  namely,  at  C, 
the  vertex  of  the  triangle,  or  in  some 
point  within  the  triangle;  and  when 
returned  to  their  original  positions,  they 
— while  making  the  prolongations  of 
CA  and  CB,  respectively  —  will  in- 
tersect at  some  other  point  C,  below 
AB.  Or,  in  other  words,  the  two  lines  CAD  and  CBE  will 
intersect  in  two  points  Cand  C,  — which  is  impossible  (The- 
orems 2,  5  and,  corollaries).  Therefore,  it  is  impossible  that 
in  the  triangle  ABC,  a  +  6  ^  2  rt.  Z's.  Q.  E.  D. 

Corollary  I.  —  If  in  the  formula  a  +  6  ^  2  rt.  Z's,  we  sepa- 
rate the  case  of  equality,  namely,  a  +  6  =  2  rt.  Z's,  we  shall  have 
a=/  ABE  and  b  =  /  DAB  ;  and  the  two  straight  lines  mak- 
ing such  angles  with  the  secant^  cannot  meet  either  below  or 
above  the  secant,  —  for,  in  either  case,  they  would  have  to 
meet  simultaneously  also  on  the  other  side  of  the  secant,  which 
is  impossible.  Therefore,  if  two  straight  lines  intersected  by 
a  third  one  make  with  it  two  interior  angles  on  the  same  side 


Fig.  16. 


99 

of  the  secant,  equal  to  two  right  angles,  they  cannot  meet,  even 
if  produced  indefinitely  both  ways. 

Corollary  II.  —  Any  exterior  angle  of  a  triangle  is  greater 
than  either  of  the  interior  and  opposite  angles. 

Because  any  of  the  interior  angles  with  its  adjacent  exterior 
angle  equals  two  right  angles  (Theorem  10)  ;  while  with  eithe  of 
the  opposite  interior  angles  it  is  less  than  two  right  angles ;  hence, 
each  of  these  last  ones  is  less  than  the  exterior  opposite  to  it. 

Corollary  III.  —  If  one  of  the  interior  angles  of  a  triangle 
is  a  right  or  an  obtuse  angle,  either  one  of  the  remaining  two  is 
less  than  a  right  angle ;  therefore,  in  any  triangle  there  can  be 
no  more  than  one  right  or  obtuse  angle,  and  not  less  than  two 
acute  angles. 

Corollary  IV.  —  A  perpendicular  is  the  shortest  line  from  a 
point  to  a  straight  line  (Proposition  XYIII  of  Euclid's  Ele- 
ments). A  perpendicular  is,  therefore,  assumed  to  represent 
the  distance  from  a  point  to  a  straight  line. 

Definition  X.  —  A  quadrilateral  is  a  figure  bounded  by  four 
sides  containing  four  interior  angles.  A  rectilinear  quadrilat- 
eral is  one  bounded  by  the  segments  of  four  straight  lines  — 
of  fixed  length  each  —  between  four  points  which  are  the  ver- 
tices of  the  quadrilateral,  —  each  vertex  being  connected  only 
with  two  adjacent  ones.  A  plane  quadrilateral  is  one  that  can 
be  placed  wholly  in  a  plane.  We  shall  have  occasion  to  use 
quadrilaterals  of  fixed  sides  only,  but  not  of  fixed  angles.  Of 
course,  such  are,  so  to  speak,  non-material  ones,  i.  e.,  bounding 
no  fixed  plane  area,  and  in  such,  the  relative  distances  of  the 
four  vertices  are  fixed  only  for  the  four  (out  of  all  six  pos- 
sible) pairs,  constituting  the  ends  of  the  four  sides  respec- 
tively. While  the  three  distances,  AB,  BCy  CA,  must 
necessarily  fix  all  distance-relations  between  three  points  —  a 
distance  being  a  relation  between  one  pair  (number  of  com- 
binations of  three  different  things  taken  two  at  a  time),  only 
six  distances  will  be  sufficient  to  fix  uniquely  the  distance- 
relations  between  four  points  (number  of  combinations  of  four 
different  things  taken  two  at  a  time)  ;  hence,  four  distances  are 
insufficient  in  the  case  of  a  quadrilateral.  If,  however,  the 
quadrilateral  is  restricted  to  lie  in  a  plane,  we  have  an  addi- 
tional relation, — and  only  one  additional  distance,  like  one  diago- 
nal (the  distance  between  either  pair  of  the  opposite  vertices), 
will  be  sufficient  to  determine  the  complete  form  of  the  quadri- 


100 


lateral.  This  property  of  a  quadrilateral  is  expressed  by  say- 
ing that  a  quadrilateral  can  "  rack."  We  shall  call  such  a  quad- 
rilateral with  variable  angles  an  immaterial  quadrilateral. 

Theorem  15. — In  any  rec- 
tilinear quadrilateral,  whether 
plane  or  not,  whose  opposite 
sides  are  equal,  the  opposite  an- 
gles are  equal. 

Demonstration. —  Let  AB  =: 
DC,  AD  =  BC.  Then  join- 
ing A  C  and  DB,  we  get 


Fig.  17. 


aABC=  a  CJDA 


and 


A  ABB  =  A  CDB ; 

.  • .  ZABC=  Z  CDA,     ZBAD  =  ZDCB. 

Q.  E.  D. 

Corollary  I.— It  follows,  that  ZABD  =  ZBDC  and  ZADB 
=  ZCBD;  similarly  ZCAB==  ZACD  and  ZACB  =  ZCAD, 
—  that  is,  the  angles  made  by  the  same  diagonal  with  the  two 
opposite  sides,  are  equal. 

Corollary  II.  —  It  also  follows,  that  in  a  plane  quadrilateral, 
with  equal  opposite  sides,  two  non-opposite  sides  and  the 
angle  enclosed  by  them  are  suf- 
ficient to  determine  the  whole 
quadrilateral. 

Theorem  16.  —  Two  plane 
quadrilaterals  ABCD  and 
A' B'  CD' ,  having  three  sides  and 
two  included  angles  in  the  one 
equal  respectively  to  the  corres- 
ponding three  sides  and  included 
angles  in  the  other,  are  equal  to 
each  other. 

Demonstration.  —  If 


DA  =  D'A\ 
AB  =  A'B', 


Fig.  18. 


and  also, 


BC=^B'C', 


101 


Osim'fi) 


0, 


Of 


X' 


ZDAB  =  ZD'A'B\ 

ZABO=  ZA'B'C, 

then,  by  superposition,  we  see  that  the  two  quadrilaterals  coin- 
cide with  each  other.  Q.  E.  D. 

Theorem  17.  —  In  a  plane  quadrilateral,  two  of  whose  op- 
posite sides  are  equal  and  in 
which  the  interior  angles  made 
by  these  with  one  of  the  re- 
maining sides,  are  supplemen-  A«(m+/) 
tary,  this  last  side  cannot  be 
greater  than  the  side  opposite 
to  it. 

Let      OA  ^  O^A^,  Z  AOO^ 
+  /  00,A^  =  2  rt.  Z  's  ; 

then  00^  >  AA^. 

For,  let  00^>AA^,  Then 
since  OA  =  O^A^  and  Z  AOO^ 
:=  Z  Afifi^j  the  supplement  of 
Z  OOyA^y  we  can  apply  the 
lower  side  of  the  quadrilateral 
to  its  upper  side  O^A^j  and  we 
get  ^,^,<  0,0,  =  00,. 

Now,  suppose  00^  —  AA^=l, 
some  length ;  then,  every  time 
0  is  transposed  a  distance  =  00^ 
along  the  straight  line  XX',  A 
is  transposed  a  distance  =  00^ 
—  L  Let  OA  =  ml  -{-  ?„  where 
m  is  a  positive  integer,  and  l^ 
is  either  zero  or  a  length  less 
than  I.  Then,  applying  this 
quadrilateral  to  itself,  along  the 
same    straight    line  XX ,    2(m  Fia.  19. 

+  1)  times,  we  get, — 

<^<^2U-f  1)=  2  (m  +  1)  00,  =  2  (m  +  1)  ^^,  +  2  (m  -h  1)  ^ 
>2(m+l)  AA,-\-2ml-{.2l,  =  AA,A,>>>A,^^^,^+  OA -\- 


^2U+l)  ^2(m+l)> 


102 


that  is,  the  straight  line  between  0  and  0^^^^  ^^  is  greater  than 
the  broken  line  between  these  same  two  points  —  which  is 
absurd  *  (Euclid  XX,  Book  1).     Hence,  00^  >  AA^. 

Q.  E.  D. 

Corollary.  —  The  sum  of  the  three  angles  of  a  triangle  can- 
not be  greater  than  two  right  angles. 

For,  if  ^50  is  such  a  triangle,  then  apply- 
ing to  it  an  equal  triangle  BCD,  where  BD 
=  ^(7  and  CD  =  AB,  we  get  ZDBC+  ZCBA 
+  ZBAC=^  ZC+  Z  B+  ZA>2rt.Z's; 
therefore,  if  Z  D'BC+  ZB+  ZA  =  2  rt. 
Z's,  and  D'B=DB,  we  have  ZD'BC< 
ZDBC,  whence  CD'  <  CD  (Euclid  I,  XV) ; 
we  have  also,  D'B  =  CA  and  ZD'BA-{- 
ZBAC=  2  rt.  Z's,  ^.  e.,  D'BACis  just  such 
B  a  plane  quadrilateral  as  has  been  discussed  in 
the  theorem,  and  CD'  <AB,  —  which  is  im- 
possible. Hence,  the  sum  of  the  three  angles 
of  a  triangle  cannot  be  greater  than  two  right 
angles.  Q.  E.  D. 

FIG.  20.  Theorem  18.  —  An  immaterial  quadrilateral 

whose  opposite  sides  are  equal,  can  so  "  rack "  or  change  its 
angles  that,  while  one  of  them  is  passing  through  all  possible 
magnitudes,  the  middle  points  of  one  pair  of  opposite  sides 
shall  always  remain  at  a  distance  from  each  other,  equal  to 
each  of  the  other  pair  of  opposite  sides. 

Demonstration. — Let  ABCD  be  a  quadrilateral  whose  oppo- 
site sides  are  equal,  and  E  and  _F,  the  mid-points  of  AD  and 
BC,  respectively. 

We  observe  first,  that,  ABCD  being  an  immaterial  quadrilat- 
eral, the  only  restriction  imposed  upon  the  relative  position 
of  its  sides  is  that  resulting  from  the  equality  of  either  pair 
of  its  opposite  sides.  In  any  deformation  or  "  racking  "  of  the 
quadrilateral,  the  sides  need  not  stay  in  one  plane,  if  this  is  re- 
quired by  the  conditions  of  the  relative  motion  to  which  they 
are  subjected,  so  that  any  pair  may  be  in  a  different  plane  from 
the  remaining  pair.  (B),  for  instance,  is  free  to  move  in  some 
path  or  other,  situated  in  a  spherical  surface  around  (C),  this 


*  See  note,  p.  116. 


103 

last  moving  in  a  sphere  about  (Z)),  and  this  one  about  (A)  — 
provided  the  four  distances  do  not  change  during  these  motions. 
Initially  the  distance  EF  is  not  given ;  it  is,  however,  capable 
of  determination.  At  least  in  one  of  the  infinite  number  of 
relative  positions  which  can  be  assumed  by  the  points  (J.),  (B), 
(C),  and  (D),  as  just  defined,  the  distance  jEJi^will  be  equal  to 
each  of  the  sides  (AB)j  (CD) ;  this  will,  evidently,  be  the  case 


when  all  four  points  (B),  (C)  (J.),  (D)  will  fall  upon  the  same 
straight  line,  say,  upon  B' ,  C,  A' ,  I)',  respectively,  where  B'  C 
=  2BF=2CF  =  A'I)'=2AE=:2EB,  and  A'B'=:AB^ 
DC^B'C. 

For  ZB  =  ZD  —  always  (Theorem  15);  hence,  when  the 
first  becomes  zero,  the  second  must  also  become  zero  ;  that  is, 
when  (BC)  falls  upon  (AB),  (AD)  falls  upon  (DC);  hence, 
(A'B')  will  be  in  the  same  straight  line  with  (B'  C)  when  and 
only  when  (A' D)  i^  in  the  same  straight  line  with  (DC), 
where  the  primed  letters  denote  some  particular  position  of  the 
sides  ;  in  other  words,  B'  will  be  in  the  same  straight  line  with 
A'  C y  and  D'  will  be  in  the  same  straight  line  with  these. 
Hence,  F  and  E  are  the  middle  points  of  B'  C  and  A' D' , 
respectively.     We  have,  then, 

EF=  EA'  +  A'B'-  FB'  =  ^A'D'  +  A'B'  -  ^C'B' 

=  A'B'  ==AB=  CD, 
since  A'D'/2  =  C'B'/2, 

In  this  position,  then,  let  (EF)  be  hinged  to  the  two  ends  of 
a  straight  line  of  fixed  length,  or  let  E  and  F  become  fixed,  so 
that  (BC)  singly  may  revolve  around  i^in  a  sphere,  and  (AD) 
singly,  around  E  in  a  sphere.  I  say  that  the  quadrilateral  may 
be  brought  out  of  coincidence  with  a  single  straight  line  and 
may  assume  all  values  for  any  one  of  its  angles,  provided  the 
equality  of  opposite  angles  is  preserved,  not  only  in  the  whole 


104 


quadrilateral,  but  also  in  the  partial  quadrilaterals.  For,  if 
not,  the  reason  of  this  must  be  sought  only  in  the  mutual 
geometrical  relations  of  the  distances,  which  are  the  only  parts 
of  our  immaterial  quadrilateral  given,  including  now  the  addi- 
tional distance  jEJi^  between  the  mid-points  of  (AB)  and  (JBC)  ; 
that  is  to  say,  these  distances  must  be  incompatible  with  an 
angle  b  other  than  zero,  so  that  an  attempt  to  change  this  value 
would  involve  a  contradiction  in  theory  and,  hence,  a  break  in 
the  distance-relations  in  practice.  Calling  the  interior  angles 
of  the  two  partial  quadrilaterals  as  in  the  figure,  we  get  six 
independent  relations  between  the  angles,  which,  on  the  suppo- 
sition that  the  motion  sought  is  possible,  must  hold  for  all 
values  of  the  angle  6,  the  value  6  =  0  included.  We  have 
also,  in  addition,  two  independent  relations,  derived  from  the 
supposition  that  (BFC)  and  (AED)  are  to  remain  straight 
lines  in  the  whole  course  of  the  motion.  We  have  thus  at  first 
appearance  relations  enough,  and  just  enough,  to  determine  each 
angle,  if  such  a  determination  is  at  all  involved  in  the  mutual 
relations  of  the  angles  and  distances.  Accordingly,  if  the  mo- 
tion postulated  were  impossible,  these  relations  would  have  to 
lead  us  to  a  contradiction  for  any  value  of  b  other  than  6  =  0. 
Now,  instead  of  leading  to  such  a  contradiction,  the  first  six 
relations  give  us,  as  an  immediate  consequence,  a  new  relation 
verified  by  the  remaining  two,  —  showing  that  our  supposition 
does  not  involve  any  contradiction,  and  that  such  a  motion  can 
be  realized.     The  relations  referred  to  are  : 


^  =  S,II. 


1)  a  =  ai 

4)y5=6' 

2)7  =  c 

.  • .  a  =  7, 1 ;         5)  3  =  cZ 

3)  a=c 

6)b==d. 

r  7)  a  +  S  =  2  rt.  Z's 
Besides                   \ 

8)  /3  +  7  =  2  rt.  Z^s. 

Combining  I  and  II,  we  get, 

a+8  =  y3+7,ni,— 

which  agrees  with  (7)  and  (8).  Of  course,  I  and  II  can  be 
realized  without  the  relations  (7)  and  (8),  when  (BFC)  and 
(AED)  are  not  restricted  to  remain  straight  lines.     But  the 


105 

argument  is  thereby  not  invalidated  —  that  the  motion  presup- 
posed in  the  theorem  cannot  be  impossible  under  the  only  con- 
ditions given.     It  may,  of  course,  be  impossible  under  certain 


Fig.  21. 

additional  conditions,  like  the  restriction  of  the  lines  to  slide  on 
certain  surfaces  ;  but  such  a  restriction  is  excluded  in  the  given 
conditions,  where  only  abstract  distances  are  given.  The  motion 
itself  may  involve  certain  positions  as  possible  and  others  as 
impossible,  —  which  circumstance  we  shall  presently  proceed  to 
investigate  only  in  so  far  as  will  be  necessary  for  our  main 
purpose,  which  is  the  establishment  of  the  theory  of  parallels. 
Remark.  —  Such  a  motion  would  also  be  possible,  even  in 
the  case  of  spherical  arcs  of  great  circles,  provided  these  arcs 
are  at  liberty  to  move  out  of  the  surface  to  which  they  belong 
initially,  —  that  is,  provided  an  additional  restriction  is  not  im- 
posed upon  them  that  every  point  of  all  five  must  remain,  dur- 
ing the  motion,  at  a  constant  distance  from  the  center  of  the 
sphere  to  which  they  belong  when  the  spherical  angle  h  is 
zero.  In  other  words,  in  order  that  such  a  motion  be  realized 
for  arcs  of  great  circles  of  the  same  sphere,  the  planes  deter- 
mined by  the  arcs  in  motion  can  all  have  a  common  point  of 
intersection  only  when  the  spherical  angle  of  two  of  these  linked 
arcs  is  zero  or  tt  —  i.  e.,  when  they  coincide  with  one  and  the 
same  great  circle  ;  for  all  other  values  of  the  angles  these 
planes  cannot  intersect  in  the  same  point,  —  in  other  words,  the 
arcs  must  leave  the  common  surface  to  which  they  originally 
belong,  as  can  easily  be  proved  when  we  come  to  a  maturer 
stage  of  the  science  of  geometry,  to  which  the  treatment  of 
spherical  geometry  properly  belongs.  A  similar  remark  holds 
with  respect  to  geodesies  upon  a  pseudosphere.  It  seems,  that 
proceeding  from  these  considerations,  it  ought  to  be  possible  to 
prove  that,  whenever  the  normals  to  a  surface-element  of  a  ho- 
mogeneous surface  meet  in  a  point,  or,  what  is  the  same  thing, 
whenever  the  surface  has  constant  positive  curvature,  there  must 


106 


be  an  excess  of  the  sum  of  the  three  angles  of  a  triangle  over 
two  right  angles ;  and  whenever  the  normals  lie  in  different 
planes,  or  the  homogeneous  surface  has  constant  negative  cur- 
vature, there  is  a  deficiency  of  the  sum  of  the  three  angles  of 
a  triangle  from  two  right  angles. 

Analytically,  the  proof  of  the  possibility  of  such  a  motion 
as  defined  in  the  theorem,  is  obtained  also  by  showing  that,  if 
we  add  to  the  above  eight  relations  an  arbitrary  relation  h—  6, 
the  determinant  of  the  nine  linear  equations  in  nine  variables  — 
when  the  equations  are  made  homogeneous  —  vanishes  identi- 
cally, proving  that  h  can  have  any  value.  The  determinant  in 
question  is  —  if  a,  a,  b,  jS,  c,  7,  d,  3,  1  is  the  order  in  which 
the  variables  are  written  —  as  follows  : 

1-10000000 


0   0  1-1 

0   0  0   0 

0   0  0   0 

10  0   0 


0  0  0  0  0 
1-10  0  0 
0  0  1-10 
10   0   0   0 


0   0 
0   1 


0   0   0 


0   0 


0   0   0   0   0   1  -TT 


0   0   0 
0   0   1 


0 


0   0  - 


=  0. 


an 


0   0   0   0   0-^ 

Q.  E.  D. 

Theorem  19.  Lemma.  —  Given 
angle  and  a  straight  line  through  its  ver- 
tex, not  in  the  plane  of  the  angle,  the  sum  of 
the  angles  made  by  this  line  with  the  sides 
of  the  angle,  is  greater  than  the  angle. 

The  theorem  needs  proof  only  for  the 
supposition  that  Z  COA  and  Z  COB, 
taken  singly,  are  each  less  than  Z  A  OB. 
Connect  any  two  points,  a  and  6,  on  the 
sides  of  the  given  angle  AOB,  by  a 
straight  line  ab ;  this  last  will  be  in  the 
plane  of  the  angle  (Theorem  12),  and 
hence  OC,  the  line  through  0  outside  the 


107 

plane,  cannot  meet  ah.  Pass  now  a  Im^Od  in  the  plane  A  OB, 
at  an  angle  aOd  equal  to  Z  ^0(7  and  cutting  ah  in  some  point 
d  —  which  it  must  do,  since  it  cannot  cut  either  Oh  or  Oa 
again  (Theorem  6  and  corollary),  and  because  the  straight  line 
from  0  will  pierce  a  sphere  inclosing  the  whole  triangle  ahO 
(Theorems  2-5  and  corollaries).  Lay  off  on  OCa  segment  Oc 
equal  to  Od,  and  join  ca,  ch  ;  then  ca  =  da  (Theorem  11),  and 
in  the  triangle  ach^  ac  -^  ch'^  ad  +  dh  (Euclid  I,  XX  *) ;  hence, 
ZcOh  >  ZdOh  (Euclid  I,  XXY).  .  •.  ZaOc  +  ZcOh> 
ZaOd+  Z  dOh,  or  ZAOC  +  I  COB>  ZAOB. 

Q.  E.  D. 

Theorem  20.  —  In  the  motion  considered  in  Theorem  18,  the 
four  sides  of  each  quadrilateral  will,  at  any  instant,  be  in  one  plane. 

For  from  I,  a  =  7,  or  from  II,  /3  ==  8,  combined  with  (7), 
a  +  3  =  2  rt.  Z's,  it  follows  that  7  -f-  8  =  2  rt.  Z's.  Now,  join- 
ing FD,  we  find  that  if  CF  is  always  in  the  plane  of  A  DFEy 
then  CD  J  FB^  FA,  and  AB  are  in  the  same  plane,  and  the  theo- 
rem is  conceded.     But  if   CF  is  sometimes  out  of  the  plane 


^ , ,_ 

n 

Id — -..>,..^^ 

cj 

f^^ 

' — ~-— ~^<J/ 

f 

gt/f 

I 

^JB 

Fig.  23. 

I) FE,  then  Z  CFF  determines  a  plane  different  from  plane  DFE 
—  that  is,  BF  is  not  in  the  plane  of  Z  CFE ;  hence,  by  preced- 
ing lemma,  Z  CFD  +  Z  DFF>  Z  CFE,  or  Z  CFD  +  Z  FDC 
>  3  (Theorem  15)  ;  and  since  0=7,  we  get  c  -\-  Z  CFD  + 
Z  FDC>  7  4-  5, or  Z  nCF+  Z  CFD-j-  Z  FDC>  2rt.  Z^s— 
which  is  impossible  (cor.  to  Theorem  17).  Hence,  CF  is  always 
in  the  plane  DFE,  and  the  theorem  is  proved  as  before. 

Q.  E.  D. 

Corollary  I.  —  It  follows,  that  any  immaterial  quadrilateral 
vnth  equal  opposite  sides,  can  move  in  a  plane  so,  that  one  of  its 
angles  assume  any  value  we  please,  and  that  the  distarwe  hetween 
the  middle  points  of  one  pair  of  opposite  sides  constantly  remain 
equal  to  each  of  the  remaining  pair  of  opposite  sides. 

*  See  also  note,  p.  97. 


108 

Corollary  II.  —  It  is  evident  that  the  sum  of  the  interior 
angles  and,  hence,  also  of  the  exterior  angles,  adjacent  to  the 
same  side  in  any  of  the  positions  of  such  a  moving  quadri- 
lateral, is  equal  to  two  right  angles,  since  2  rt.  Z's  =  a  -f-  /^  = 
y-^8  =  d-\-c=a-\-h  —  b-\-c,  etc. 

Theorem  21.  —  The  sum  of  the  interior  angles  adjacent  to 
the  same  side  of  any  plane  quadrilateral  with  equal  opposite 
sides,  is  equal  to  two  right  angles. 

For,  an  immaterial  quadrilateral  having  sides  of  equal 
length  with  the  given  one,  can  assume  such  a  position  in  a 
plane,  that  the  angle  enclosed  between  any  two  of  its  non-oppo- 
site sides,  be  equal  to  the  corresponding  angle  in  the  given 
quadrilateral.  And  as  the  sum  of  the  interior  angles  adjacent 
to  the  same  side  of  this  immaterial  quadrilateral  in  the  particu- 
lar position,  is  equal  to  two  right  angles,  it  follows  by  Corollary 
II  to  Theorem  15,  that  the  sum  of  the  interior  angles  adjacent 
to  the  same  side  in  our  given  quadrilateral,  is  likewise  equal  to 
two  right  angles.  Q.  E.  D. 

Corollary  I.  —  The  sum  of  the  three  interior  angles  of  any 
triangle  is  equal  to  two  right  angles. 

For,  if  in  the  triangle  ABC  we  produce  AB  to  E,  Z  CBE> 
C;  hence,  making  Z  CBD  =  ZBCA^  BD  falls  within  angle 
CBE  (Theorem  14,  Cor.  II).     Taking  BD  =^  AC,  and  joining 

CD,  we  have,  A  CBD  =  a  BCA, 
and  CD  =  AB  (Theorem  11), 
hence,  we  have  a  plane  quadri- 
lateral with  equal  opposite  sides 
therefore,  Z  CAB  -h  Z  ABD  = 
2  rt.  Z's,  or  ZA  +  ZB  +  ZG 
=  2  rt.  Z's. 

Corollary  II.  —  The  exterior 
angle  of  a  triangle  is  equal  to  the  two  interior  and  opposite 
angles. 

Corollary  III.  —  The  four  angles  in  any  plane  quadrilateral 
equal  4  rt.  Z^s,  since  it  can  be  divided  into  two  triangles  having 
their  six  angles  coincident  with  the  four  given  ones. 

Theorem  22.  —  Two  straight  lines  perpendicular  to  a  third 
one  in  the  same  plane  with  it,  — 

1)  Will  both  be  perpendicular  to  any  other  perpendicular 
drawn  from  any  point  in  the  one  to  the  other ; 


109 

2)  Will  both  have  the  same  inclination  towards  any  secant 
to  both ; 

3)  Will  be  everywhere  equidistant  from  each  other,  i,  e.,  any 
perpendicular  from  one  to  the  other  will  be  of  the  same  length 
with  any  other. 

Demonstration.  —  1)  Let  J.j5,  CD  be  both  perpendicular  to 
the  same  straight  line  KL  in  the  same  plane ;  and  draw  any 
other  perpendicular  UN,  from  any  other  point  31  in  AB  to 
CD ;  and  join  LM.  The  sums  of  the  interior  angles  of  each  of 
the  two  triangles  are  equal  to  two  right  angles  singly,  and  to- 


FiG.  25. 

gether  they  are  equal  to  four  right  angles.  But  the  six  angles 
of  both  triangles  make  up  together  the  four  angles  of  the  quad- 
rilateral KLMN.  Of  these  last,  the  angles  K,  L,  and  N  are 
each  a  right  angle  (by  construction) ;  therefore,  Z  31  also  is 
a  right  angle,  and  hence  any  perpendicular  to  CD,  from  any 
point  in  AB,  will  also  be  perpendicular  to  AB.  For  the  same 
reason,  any  perpendicular  to  ABj  from  any  point  in  CD,  will 
also  be  perpendicular  to  CD. 

2)  In  the  same  diagram,  if  P3fL  is  a  secant  toAB  and  CD, 
which  are  both  perpendicular  to  a  third  line  in  the  same  plane, 
then  draw  from  L  a  perpendicular  LK  to  AB,  and  from  Jf,  a  per- 
pendicular 3fN  to  CD.  The  first  will  also  be  perpendicular  to 
CD  J  and  the  second,  to  AB  (section  1  of  our  proposition).  Now, 
in  the  triangle  L3fN  we  have  Z  L3IN -{-  Z  NL3I  =  2  rt.  Z's  — 
Z  N=  rt.  Z  (corollary  to  Theorem  21);  besides,  /.L3IN-\- 
Z  LiOr=  31=  rt.  Z.  Hence,  Z  L3^1N  -h  /_  NL3f=:  Z  L3fN-{- 
Z  L3IK;  Z  NL3I=  Z  L3IK,  or  Z  PLD  =  Z  P3IB  (Theorem 
10).  That  is,  both  AB  and  CD  have  the  same  inclination 
o  wards  any  common  secant. 


110 

3)  If  KL,  MN  are  any  two  perpendiculars  to  both  AB  and 
CDy  which  are  situated  in  the  same  plane,  and  ML  joins  the 
opposite  angles  M  and  X,  then  Z  NLM  =  Z  LMK ;  Z  LMN 
=  Z  MLK  (section  2  of  our  proposition),  and  ML  is  common 
to  both  triangles  ;  therefore,  A  L3IN=  A  MLK,  and  MN= 
KLj  —  and  as  the  same  reasoning  applies  equally  to  any  other 
two  perpendiculars  to  both,  all  the  points  of  either  line  AB  ov 
CDy  are  at  equal  distances  from  the  other. 

Definition  XI.  —  Two  straight  lines  perpendicular  to  a  third 
in  the  same  plane,  as  having  the  same  inclination  towards  any 
common  secant  and  being  everywhere  equidistant  from  each 
other,  are  said  to  be  parallel  to  each  other,  meaning  —  beside 
each  other,  or  going  in  the  same  direction  and  being  everywhere 
at  the  same  distance  from  each  other. 

Corollary  I.  —  Any  perpendicular  to  one  of  a  pair  of  parallel 
lines,  in  the  same  plane  with  both,  if  produced  indefinitely 
must  meet  the  other  at  right  angles. 

For,  if  from  the  point  of  intersection  with  the  first  a  perpen- 
dicular be  drawn  to  the  other,  it  must  also  be  perpendicular  to 
the  first  (section  1)  and,  therefore,  coincide  with  the  perpen- 
dicular to  the  same,  previously  drawn. 

Corollary  II.  —  From  any  given  point  without  a  given 
straight  line  only  one  parallel  can  be  drawn,  namely,  that  line 
which  is  at  right  angles  to  the  perpendicular  from  the  point  to 
the  given  line. 

Corollary  III.  —  Any  two  points  at  the  same  distance  from 
a  given  straight  line,  in  the  same  plane  with,  and  on  the  same 
side  of  it,  determine  another  straight  line  parallel  to  the  first. 
For,  joining  these  points  and  drawing  the  perpendiculars  to  the 
given  straight  line,  we  get  a  plane  quadrilateral  which  is  con- 
gruent with  the  duplicate  turned  over,  so  that  the  equal  perpen- 
diculars become  interchanged  in  position  ;  hence,  the  angles  op- 
posite the  given  right  angles  are  equal,  and  as  their  sum  equals 
two  right  angles  (Theorem  19,  corollary  III),  each  is  equal  to 
one  right  angle  ;  that  is,  the  line  connecting  the  points,  and  the 
given  line,  are  both  perpendicular  to  the  same  straight  line  in 
the  same  plane  with  these.     Hence,  they  are  parallel. 

Corollary  IV.  Any  two  straight  lines  lying  in  the  same 
plane  and  having  same  inclinations  towards  a  common  secant 
are  parallel ;  for  if  a  parallel  to  one  of  the  given  lines  is  con- 
structed through  the  intersection  of  the  other  with  the  secant, 
the  corollary  becomes  evident  (section  2). 


Ill 

Theorem  23.  —  Not  more  than  two  points  of  equal  distances 
from  a  given  straight  line,  can  be  situated  in  another  straight 
line  which  is  in  a  different  plane  from  that  passing  through  the 
given  line  and  one  of  these  points. 

Demonstration.  —  Let  AB^  CE  be  the  two  straight  lines,  of 
which  EC  is  not  in  the  plane  BA  C  passing  through  the  line 
AB  and  the  point  (7;  and  let  the  distances  of  the  points  C 
and  E  from  AB  (i,  e.,  the  perpendiculars  EB  and  CA  drawn 
from  these  points  to  AB)  be  equal.  Then  no  other  point  on 
EC  can  be  of  the  same  distance  from  AB. 

For,  produce  the  plane  BAG  indefinitely  beyond  BG,  and 
draw  initGDLAGy.'.W  AB  (Definition  XI),  and  BD 1  AB. 
BD  will  meet  CD,  and  BI)  =  AG=  BE  (Theorem  22,  Cor. 
I  and  Definition  XI).  Join  AD,  BG,  EA.  The  right-angled 
triangles  BAE^  ABB,  and  j5J.(7are  equal  (Theorem  11)  ;  there- 
fore, EA  =  J)A=^  GB,  and  Z  EAB  =  Z  DAB  =  Z  GBA. 
Also,  since  EA  is  not  in  the  plane  GAB,  Z  GAE  -f  Z  EAB 
>  Z  GAB  (Theorem  19),  or  Z  GAE  +  Z  EAB  >  Z  GAD 
-^ZDAB;  hence,  ZGAE>  Z  GAD.  Now  the  triangle 
EAG  must  have  Z  EGA  <  rt.  Z.  For,  if  we  imagine  it  re- 
moved from  its  original  position  and  applied  to  a  DA  G  in 
such  a  way  that  EA  shall  coin- 
cide with  its  equal  DA  —  the 
other  side,  inclosing  the  greater 
angle  EAG,  will  fall  without  y 

the  smaller   angle  DAG  and         / 

will  take   the   position  AG',        I         y  >>^ j _-^b 

as  in  the  diagram ;  and  join-     c'^ 
ing  GG ',  we  obtain  an  isosceles  ^         p^^  26        ^ 

triangle  AGG' ,  of  which  the 

angles  (7. and  C,  at  the  base,  are  equal,  and  each  less  than 
a  right  angle  (Cor.  to  Theorem  11  and  Cor.  Ill  to  Theorem 
14).  Then,  DG  and  GG'  form  an  angle  DGG'  <  2  rt.  Z's, 
having  its  opening  towards  A  ;  that  is,  the  prolongation  oi  G'  G 
is  separated  from  A  and  from  any  point  on  AD  by  the  half-ray 
CD.  Therefore  the  half-ray  G'  (7,  including  its  prolongation,  in 
passing  continuously,  not  through  A,  to  the  position  C'  D  contain- 
ing one  point  of  the  segment  AD,  must  first  pass  through  some 
point  on  the  prolongation  o^  AD  before  reaching  its  final  posi- 
tion. Hence  Z  A  G' D  is  less  than  some  angle  which  is  less  than 
Z  ^(7'C(Scholium  to  Theorem  12);  or  ZAG'D<ZAG'G< 


112 


UV  yiB 


Fig.  26. 


rt.  Z.  Now,  any  point  on  the  prolongation  of  EC  to  the 
left,  must  also  be  without  the  plane  DC  A,  since  one  part  of 
EC  cannot  be  without,  and  the  other  within,  the  plane  DC  A 
(Theorem  12) ;  and  any  straight  line  connecting  C  and  an- 
other point  of  the  same  distance  from  AB,  without  the  plane 
DC  Ay  makes  with  CA  an  angle  <  rt.  Z  and,  consequently, 
cannot  be  the  prolongation  of  EC,  since  such  a  prolongation 
forms  with  CA  an  angle  >  rt.  Z  (Theorem  10).  Therefore, 
no  point  of  the  same  distance  from  AB  2^  C,  can  be  on  the 
prolongation  of  EC  to  the  left.  For  a  similar  reason,  no  point 
of  the  same  distance  from  AB  as  the  two  given  points,  can  be 

on  the  prolongation  of  CE  to 
the  right,  because  CE  is  not 
in  the  plane  AEB  passing 
through  AB  and  the  point  E, 
Neither  can  a  point  of  an 
^E  equal  distance  from  AB  be 
situated  on  CE,  between  C 
and  E,  since  then,  either  E 
or  C  would  be  on  the  prolong- 
ation of  a  straight  line  connecting  two  points  of  equal  dis- 
tances from  another  straight  line — one  of  the  points  connected 
being  in  a  different  plane  from  that  passing  through  the  other 
point  and  the  other  straight  line,  — which  has  just  been  demon- 
strated to  be  impossible.  Therefore,  no  other  point  besides  E 
and  (7,  on  the  same  straight  line  with  them,  or  on  its  prolonga- 
tions, is  possible,  of  the  same  distance  from  ^j5  as  ^  and  C. 

Q.  E.  D. 

Corollary  I.  —  Hence,  any  straight  line  in  space  which  con- 
nects three  points  at  equal  distances  from  another  straight  line, 
must  lie  wholly  in  the  plane  passing  through  the  other  line 
and  one  of  its  own  points,  —  and,  therefore,  is  parallel  to  the 
other ;  and  all  that  is  said  in  Theorem  22,  with  reference  to  a 
parallel  straight  line,  is  also  true  of  any  straight  line  in  space 
having,  at  least,  three  points  at  equal  distances  from  a  given 
straight  line. 

Corollary  II.  —  It  also  follows,  that  if  two  equal  lines  inter- 
sect two  others  in  four  points,  and  three  of  these  lines  are  at 
right  angles  to  one  another,  all  four  are  in  the  same  plane,  and, 
therefore,  are  parallel,  each  pair  singly. 

Corollary  III.  —  A  parallel  can  also  be  defined  as  the  locus 


113 


o/  all  points  equidistant  from  a  straight  line,  and  coUinear  vnth 
two  given  points. 

The  definition  of  alternate  interior  and  exterior  angles  of  a 
line  intersecting  two  others  in  the  same  plane,  is  as  usually- 
given.  If  the  definition  of  parallels  is  taken  provisionally 
in  the  sense  in  which  we  have  defined  them,  we  see  that  we 
have  proved  propositions  XXVII,  XXVIII,  XXIX  of  Eu- 
clid, also  XXXII  and  some  others,  from  which  a  number  of 
corollaries  can  be  drawn,  regarding  the  conditions  of  paral- 
lelism of  straight  lines  in  space,  which  we  will  not  give 
here.  We  are  now  in  a  position  to  prove  propositions  XXX, 
XXXIII,  and  XXXIV  of  Euclid,  using  his  proofs  word  for 
word.  This  will  now  enable  us  to  deduce  Euclid's  famous 
Eleventh  Axiom,  and  thereby  extend  the  definition  of  paral- 
lelism to  any  two  lines  situated  in  the  same  plane  and  not  meet- 
ing each  other  at  a  finite  distance. 

Theorem  24.— 
Two  straight  lines 
in  the  same  plane, 
of  which  one  is 
perpendicular  to  a 
third  one,  and  the 
other  makes  an 
acute  angle  with 
it,  if  sufficiently 
produced  on  that 
side  of  the  secant 
where  the  acute 
angle  is  situated, 
must  meet  each 
other  somewhere. 

Let  5  C  be  per- 
pendicular to  AB, 
and  AE  make  an 

acute   angle    with    ^^ 

AB    at   A,    then 

AE  and    BC,   if 

produced  towards  the  opening  of  the  acute  angle,  will  meet  in 

some  point/. 

Demonstration.  Take  some  point  a  on  AE,  draw  a  perpen- 
dicular from  it  to  A B  (Euclid,  prop.  XI).     This  perpendicular 


I  m  n  pBq 

Fig.  27. 


114 


will  cut  AB  in  some  point  I  between  A  and  JB,  and  not  on  its 
prolongation  AR ;  otherwise  ZBAE,  being  according  to  sup- 
position an  acute  angle,  would  be  greater  than  a  right  angle 
(Theorem  14,  Cor.  II)  —  which  is  absurd.  Let  now  Al  be  con. 
tained  in  AB  m  times  (m  being  a  whole  number),  or  m  times 
with  some  remainder  less  than  AL  Take  upon  AEy  beginning 
from  A  and  proceeding  towards  -£J,  m  +  1  parts  equal  to  Aa, 
namely  Aa,  ab,  6c  .  •  •,  and  let  /  be  the  end  of  the  last  part. 


A  Im  n  pBq 

Fig.  27. 


Draw  now  perpendiculars  aa^,  hh^,  cc^-  •  -ff^  to  AD  which  is  made 
perpendicular  to  AB  (Eucl.,  prop.  XI) ;  they  will  also  be 
parallel  to  AB  and  to  one  another  (Defin.  XI).  Draw  also 
6m,  en  J  •  •  -/g,  parallel  to  BC,  AD,  al,  —  which  will  all  meet  at 
right  angles  all  the  perpendiculars  to  AD  (Cor.  1  to  Theorem  22). 
Let  the  vertices  of  these  right  angles  be  g,  A,  ^,  •  •  •/.  All  these 
perpendiculars  will  represent  two  sets  of  parallels.  The  triangles 
Aaly  ahg,  ---efk  will  be  equal  to  one  another  because  they  have 
one  side  and  two  adjacent  angles  equal  respectively,  in  all 
of  them  —  those  adjacent  angles  being  alternate  angles  (Theorem 


115 

22,  section  2,  and  Euclid,  prop.  XXYI).  Therefore  Al  = 
ag  —  hh-  "  ek,  and  therefore  also  —  according  to  Euclid,  proposi- 
tion XXXIII — Al  =  lm  =  mn  •  .  •  =  pq.  For  the  same  reason 
a^a  =  Aly  bj)  =  Am,  c^c  =  An  •  •  'f^f==  Aq.  But  Aq,  contain- 
ing m  -f  1  times  Al  is  greater  than  AB  ;  hence  f^f>AB,  In 
other  words,  the  distance  of  f  from  AD  is  greater  than  the  dis- 
tance between  the  parallels  BC,  AD,  which  is  everywhere  the 
same  and  equal  to  AB  ;  hence,  y  must  lie  beyond  the  space  in- 
closed between  both  the  parallels  —  and  since  it  is  a  point  on 
AE,  AE  must  intersect  ^Cin  some  point  of/'  below/. 

Q.  E.  D. 

Corollary  I.  —  Two  straight  lines  in  the  same  plane,  which 
cannot  meet  how  far  soever  produced  both  ways,  are  parallel 
to  each  other.  For,  any  perpendicular  to  one  of  them,  drawn 
from  any  point  in  the  other,  cannot  make  an  oblique  angle  with 
the  latter  —  otherwise  they  would  meet  on  the  side  of  the  acute 
angle ;  it  must,  therefore,  be  at  right  angles  to  both  —  or  both 
the  lines  must  be  parallel  (Theorem  22,  Defin.  XI). 

Q.  E.  D. 

Corollary  II.  —  Any  straight  line  intersecting  one  of  a  pair 
of  parallel  lines,  and  in  one  plane  with  the  other,  if  produced 
sufficiently  far,  must  meet  the  latter. 

For,  being  in  one  plane  with  both,  were  it  not  to  meet  the 
second  somewhere,  it  would  be  parallel  to  it  (preceding  Corol- 
lary) and  hence  also  to  the  first  (Euclid,  prop.  XXX)  ;  but  it  is 
not,  since  it  intersects  the  first.  Therefore  it  must  likewise 
intersect  the  other,  somewhere  at  a  finite  distance.   Q.  E.  D. 

With  this,  the  theory  of  parallels,  as  well  as  that  of  the  elements 
of  geometrical  measurement  —  distance,  straight  line,  plane, 
angle,  circle,  etc.,  —  is  firmly  established.     We  see,  then,  that 

THE  FOUNDATIONS  OF  THE    EUCLIDIAN   GEOMETRY   rest   OU   a 

much  firmer  basis  than  mere  arbitrary  assumptions  verified  by 
experience  to  a  very  great  degree  of  approximation.  They 
are,  rather,  implanted  in  the  very  nature  of  our  logic,  being  to 
a  great  degree  what  the  Kantists  call  a  priori,  and  are  empirical 
only  in  so  much  as  all  our  conceptions  of  quantity,  form,  and 
motion  depend  upon  experience. 


AUTOBIOGEAPHY. 

The  author  of  this  DissertatioD,  Israel  Euclid  Rabinovitch, 
was  born  in  the  month  of  June  of  the  year  1861,  in  the  town 
of  Berditchev,  Kiev  Government,  Russia.  He  obtained  his 
elementary  education  in  an  old-fashioned  rabbinical  school,  and 
afterwards  prepared  himself  by  self-instruction  for  entrance 
into  a  Russian  University,  by  having  completed  a  course  of  a 
Russian  classical  gymnasium.  Having,  however,  found  it  diffi- 
cult to  obtain  admission  there,  he  left  in  1887  for  the  United 
States,  with  the  express  purpose  of  eventually  pursuing  an 
academic  course  of  studies  in  one  of  the  American  universities. 
He  entered  the  University  of  Pennsylvania  in  1891  and,  hold- 
ing a  scholarship  there,  studied  the  elements  of  mathematics, 
including  some  engineering  courses,  under  Professors  Crawley, 
Fisher,  Barker,  and  Spangler,  during  1891-1892  and  part  of 
1892-1893,  when  he  left  on  account  of  illness.  During  the 
academic  year  of  1894-1895  he  attended  special  courses  of 
mathematics  and  German  in  Harvard  University.  In  1896 
he  entered  the  Johns  Hopkins  University,  as  a  student  in  ad- 
vanced standing,  and  has  since  been  pursuing  courses  in  mathe- 
matics under  Professors  Morley,  Craig,  Chessin,  and  Hulburt, 
and  Dr.  Cohen,  and  in  physics  under  Professor  Ames  and  Dr. 
Bliss,  and  in  philosophy  under  Professor  Griffin.  He  was  made 
a  candidate  for  the  degree  of  Doctor  of  Philosophy  in  1899, 
having  selected  Mathematics  as  his  principal  subject,  and 
Physics  and  History  of  Philosophy  as  first  and  second  subordi- 
nates, and  was  appointed  Fellow  in  Mathematics  in  June,  1900. 


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